Fletcher ran on to Derek Riordan's clever reverse pass to ease the pressure on manager Mixu Paatelainen.
Hamilton had enjoyed the better of the first half but they failed to seriously trouble Hibs keeper Grzegorz Szamotulski.
This was Hibernian's third win over Hamilton this season but the First Division champions had gone into the game as favourites.
Rangers had ended Hamilton's six-match winning home run on Saturday while Hibs had won just once in their previous 10 matches.
But Hibs looked more dangerous as the game wore on and Fletcher atoned for a bad miss to seal a much-needed win.
Both managers made several changes after being hit with selection problems.
Hamilton were without midfielder James McArthur and defenders Martin Canning and Trent McClenehan while Hibs skipper Rob Jones was absent, forcing Sol Bamba to move to centre-back.
James McCarthy was at the hub of Hamilton's promising start. The teenager outpaced Hibs right-back Darren McCormack but Szamotulski caught his attempted cutback, and soon shot narrowly wide from 20 yards.
Paul McGowan's deflected cross then clipped the crossbar after Hibs failed to deal with a long free-kick into their penalty area.
Bamba had Hibernian's first shot at goal but his long-range strike almost cleared the 30ft scaffolding behind the goal.
Chris Hogg came much closer from a similar distance as Hibs, who had right-winger Alan O'Brien supporting a three-pronged attack, began to push Hamilton back.
But the hosts soon reasserted their dominance. Jimmy Gibson had a powerful 25-yard drive saved and David Elebert hooked over with his back to goal.
Their best chance of the half fell to Kenny Deuchar in the 38th minute after Simon Mensing had headed on Alex Neil's free-kick, but the striker's half-volley drifted just wide with Szamotulski scrambling.
Denes Rosa was booked for an outrageous penalty-box dive but Hibs almost snatched the opener on the counter-attack, O'Brien stabbing Colin Nish's return pass inches wide at full stretch.
The second half started in low-key fashion but Hibs survived a scare in the 55th minute when McCarthy drove the ball across the face of goal just out of reach of two team-mates.
Steven Thicot came on for Hibs after McCormack hurt himself when he slid into the advertising boards, and the visitors immediately missed two great chances in two minutes.
Nish headed John Rankin's cross over from six yards before Fletcher sliced wide from 15 yards after Rosa had played him clean through.
Hamilton had lost their way as an attacking force but Richard Offiong almost made an instant impact after replacing Deuchar in the 65th minute. The striker slipped the ball past Thicot but scuffed his left-foot shot wide of the far post.
The game became more stretched as it entered the final 20 minutes. Fletcher shot over after making space 20 yards out and the unmarked Offiong headed weakly at Szamotulski from McGowan's cross.
But Hamilton were made to pay for switching off at an 80th-minute throw-in. Riordan controlled the ball inside the box with his back to goal and turned it inside for Fletcher to run in and finish from close range. Riordan set up Nish with another clever pass moments later but Tomas Cerny got down well to save.
Hamilton rallied but Elebert and Chris Swailes could not make headers count and Bamba just turned Brian Easton's driven cross behind as sub Rocco Quinn waited to pounce.