Rangers winger Ryan Kent gave the visitors a first half lead
Two penalties and a goal from Alfredo Morelos in the final 10 minutes gave Steven Gerrard his first Premiership away win as Rangers manager.
The win also moved them back to five points behind leaders Hearts ahead of a huge week for the Ibrox side.
Ryan Kent scored his third of the season, but Steven Boyd looked to have earned Accies a point.
James Tavernier stepped up twice in two minutes to score from the spot and Morelos added a late fourth.
A third consecutive victory means Rangers have now won on their last seven visits to Hamilton and move up to fifth in the table.
They face Spartak Moscow at Ibrox on Thursday in the Europa League before a League Cup semi-final with Aberdeen at Hampden Park on Sunday.
After a third straight defeat, Martin Canning's ninth-placed side are left with just six points from their opening nine league games.
Rangers leave it late
With Rangers without their main two strikers, Morelos and Kyle Lafferty, for next week's semi-final, the focus was on how others in the team could step up and show their potency in front of goal.
The visitors also had to show they could win away from Ibrox domestically, something they had not managed all season.
Kent and Ovie Ejaria, in particular, will be big players at the national stadium as manager Gerrard attempts to lead his team into their first cup final under his watch.
It took 31 minutes though for home Gary Woods to be forced into a meaningful save, the goalkeeper diving low to claim Ejaria's curling effort.
But Gerrard's side got their noses in front when Hamilton coughed up possession in midfield. Morelos played a delightful ball through the middle to Kent, who coolly lobbed Woods.
Woods had to tip over another Ejaria effort, but Hamilton were being left to rue missed chances of their own.
With the scores still at 0-0, Scott Martin and Darian Mackinnon both failed to hit the target after gliding in at the back post, while Aaron McGowan headed over from close range.
Even after they fell behind, Mackinnon spurned another glorious chance when he shot over after goalkeeper Allan McGregor had punched the ball into his path.
Rakish Bingham stung the palms of McGregor before James Keatings shot the rebound wide. But, as the hard luck stories were being written, they equalised in stunning fashion.
Boyd was given time and space on the edge of the box and he cracked an effort into the top corner.
However, they still had 10 minutes to hold out. Within six, they had conceded twice; referee Andrew Dallas pointing to the spot after Alex Penny's handball then after Morelos fell to the turf from a tug on his shirt.
With the final kick of the ball, Morelos blasted in with the Hamilton defence all at sea - his 12th of the season and his fourth in three games.
Win does not hide the cracks - analysis
Willie Miller on BBC Radio Sportsound
"The three goals scored without the penalties were brilliant. No way could you have predicted that Rangers were going to tuck away four at the end of the day.
"It's madness, absolutely craziness from the centre-back Alex Penny. Not dealing with the cross and then getting himself all tangled with Morelos.
"But, for long periods it was dire stuff from Rangers. It wasn't inspiring. The tempo was low, they weren't creating chances,
"It's a magnificent 4-1 victory but I don't think it hides the cracks. It was very similar this game to the Livingston game [1-0 defeat]."
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Line-ups
Match Stats
Live Text
Line-ups
Hamilton
1Woods
17Penny
5Tshiembe
4GordonSubstituted forBoydat 76'minutesBooked at 89mins
2McGowan
25MartinBooked at 18mins
18MacKinnonBooked at 75mins
7Imrie
3McMann
11MillerBooked at 12minsSubstituted forKeatingsat 69'minutes
31BrustadSubstituted forBinghamat 63'minutes
Substitutes
9Bingham
10Keatings
21Want
23Fulton
28Smith
30Boyd
46Sowah
Rangers
1McGregor
2Tavernier
6Goldson
19KaticSubstituted forWorrallat 79'minutes
16HallidayBooked at 90mins
23Coulibaly
18RossiterBooked at 72minsSubstituted forGrezdaat 82'minutes
10Ejaria
21CandeiasBooked at 43mins
20MorelosBooked at 75mins
14KentBooked at 41minsSubstituted forMcCrorieat 87'minutes
Substitutes
3Worrall
9Sadiq
13Foderingham
15Flanagan
17McCrorie
35Grezda
40Middleton
Referee:
Andrew Dallas
Attendance:
5,013
Match Stats
Home TeamHamiltonAway TeamRangers
Possession
Home41%
Away59%
Shots
Home9
Away11
Shots on Target
Home3
Away6
Corners
Home1
Away5
Fouls
Home15
Away9
Live Text
Posted at
Match ends, Hamilton Academical 1, Rangers 4.
Full TimePosted at 90'+5'
Second Half ends, Hamilton Academical 1, Rangers 4.
Goal!Posted at 90'+4'
Goal! Hamilton Academical 1, Rangers 4. Alfredo Morelos (Rangers) left footed shot from the centre of the box to the high centre of the goal.
Posted at 90'+3'
Foul by Ross McCrorie (Rangers).
Posted at 90'+3'
Steven Boyd (Hamilton Academical) wins a free kick in the defensive half.
BookingPosted at 90'+1'
Andy Halliday (Rangers) is shown the yellow card for a bad foul.
Posted at 90'+1'
Hand ball by Andy Halliday (Rangers).
Posted at 90'
Delay over. They are ready to continue.
Posted at 89'
Delay in match Rakish Bingham (Hamilton Academical) because of an injury.
BookingPosted at 89'
Steven Boyd (Hamilton Academical) is shown the yellow card.
SubstitutionPosted at 87'
Substitution, Rangers. Ross McCrorie replaces Ryan Kent.
Goal!Posted at 87'
Goal! Hamilton Academical 1, Rangers 3. James Tavernier (Rangers) converts the penalty with a right footed shot to the centre of the goal.
Posted at 86'
Penalty Rangers. Alfredo Morelos draws a foul in the penalty area.
Posted at 86'
Penalty conceded by Alex Penny (Hamilton Academical) after a foul in the penalty area.
Goal!Posted at 84'
Goal! Hamilton Academical 1, Rangers 2. James Tavernier (Rangers) converts the penalty with a right footed shot to the top left corner.
Posted at 84'
Penalty conceded by Alex Penny (Hamilton Academical) with a hand ball in the penalty area.
Posted at 82'
Delay in match Alfredo Morelos (Rangers) because of an injury.
SubstitutionPosted at 82'
Substitution, Rangers. Eros Grezda replaces Jordan Rossiter.
Goal!Posted at 80'
Goal! Hamilton Academical 1, Rangers 1. Steven Boyd (Hamilton Academical) left footed shot from outside the box to the top left corner.
Posted at 80'
Attempt missed. James Tavernier (Rangers) right footed shot from outside the box is close, but misses to the left.
SubstitutionPosted at 79'
Substitution, Rangers. Joe Worrall replaces Nikola Katic.
Posted at 78'
Foul by Scott McMann (Hamilton Academical).
Posted at 78'
Ryan Kent (Rangers) wins a free kick in the defensive half.
Posted at 77'
Attempt missed. James Keatings (Hamilton Academical) right footed shot from the centre of the box is close, but misses to the right.
Posted at 77'
Attempt saved. Rakish Bingham (Hamilton Academical) left footed shot from outside the box is saved in the centre of the goal.
SubstitutionPosted at 76'
Substitution, Hamilton Academical. Steven Boyd replaces Ziggy Gordon.
BookingPosted at 75'
Darian MacKinnon (Hamilton Academical) is shown the yellow card.
BookingPosted at 75'
Alfredo Morelos (Rangers) is shown the yellow card for a bad foul.
Posted at 75'
Darian MacKinnon (Hamilton Academical) wins a free kick in the attacking half.
Posted at 75'
Foul by Alfredo Morelos (Rangers).
BookingPosted at 72'
Jordan Rossiter (Rangers) is shown the yellow card for a bad foul.
Posted at 71'
James Keatings (Hamilton Academical) wins a free kick in the defensive half.
Posted at 71'
Foul by Jordan Rossiter (Rangers).
SubstitutionPosted at 69'
Substitution, Hamilton Academical. James Keatings replaces Mickel Miller because of an injury.
Posted at 68'
Delay in match Mickel Miller (Hamilton Academical) because of an injury.
Posted at 68'
Corner, Rangers. Conceded by Alex Penny.
Posted at 66'
Foul by Scott McMann (Hamilton Academical).
Posted at 66'
James Tavernier (Rangers) wins a free kick in the attacking half.
SubstitutionPosted at 63'
Substitution, Hamilton Academical. Rakish Bingham replaces Fredrik Brustad.
Posted at 62'
Attempt missed. Alfredo Morelos (Rangers) header from the centre of the box is just a bit too high.
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Source: bbc.com