Substitute Jasper coolly slotted home a low strike in the 88th minute for the hosts from close range, after Brendan Wiredu had burst forward and picked him out in the area.
The closest either team came to scoring before half-time was on the half hour mark when Will Smith’s header from George Thomson’s corner flew straight at Colchester keeper Shamal George.
Harrogate came close to breaking the deadlock in the 65th minute.
Jack Diamond showed great skill to leave Junior Tchamadeu on the floor before picking out substitute Lloyd Kerry, whose close-range shot crashed against the underside of the crossbar before eventually going to safety.
Colchester came close to scoring when Frank Nouble fired wide after Jasper had helped on Freddie Sears’ pass in the area.
Jasper’s fine free-kick rattled the crossbar in the 86th minute but he duly grabbed Colchester’s winner two minutes later.
Elsewhere, substitute Brett Pitman snatched a last-minute equaliser to earn Bristol Rovers a 2-2 draw at Bradford.
The game was about to move into added time when Pitman rose to meet Antony Evans’ cross and head the ball over the head of goalkeeper Richard O’Donnell.
Until then Bradford had looked set for their second successive home win.
They dominated the first half and, after keeper James Belshaw blocked Levi Sutton’s shot with his legs, deservedly went in front in the 43rd minute.
Belshaw did well to save defender Yann Songo’o’s header from Callum Cooke’s right-wing corner and Charles Vernam followed up to score from the rebound.
The visitors equalised three minutes after half-time when a corner from the left found Alfie Kilgour unmarked in the area and he headed home.
Harrogate's defeat meant Forest Green extended their lead at the top of the League Two table to four points with victory over troubled Scunthorpe.
The Iron sit bottom of the table and, a week after their 6-1 hammering by Harrogate, some fans protested outside the stadium against owner Peter Swann.
They kept out Forest Green for an hour but Jamille Matt picked himself up to convert a penalty after being fouled by Jai Rowe and Ebou Adams made it 2-0 five minutes later.
Port Vale are up to third thanks to a 3-1 victory over Barrow. The visitors took the lead through Ollie Banks in the 33rd minute but Dennis Politic equalised six minutes after the break and an own goal from James Jones five minutes later completed the turnaround before Malvind Benning added a third late on.
Fourth-placed Tranmere have struggled away from home but they secured their first win on the road – and first goal – with a 1-0 victory over struggling Carlisle thanks to Rod McDonald’s own goal.
Sutton’s good start to Football League life continued with a 1-0 win at Crawley – Isaac Olaofe scoring the only goal in the 84th minute.
Swindon slipped down to sixth despite salvaging a 2-2 draw against Rochdale.
They took the lead through Ben Gladwin in the 37th minute but George Broadbent levelled in the 57th minute and Liam Kelly looked to have grabbed the winner only for Jonny Williams to pop up in injury time and net his first goal for Swindon.
It was a big day for Oldham down the bottom of the table, with Davis Keillor-Dunn scoring an early double and Dylan Bahamboula adding a third in the second half in a 3-0 victory over Stevenage.
Exeter came from behind twice in a 2-2 draw against Newport. Matt Jay cancelled out Robbie Willmott’s opener but Courtney Baker-Richardson put County back ahead before the break.
Sam Nombe scored for the fifth successive game in the 55th minute, though, to earn the Grecians a point.
Aaron McGowan and Fraser Horsfall scored for Northampton in a 2-0 victory over Mansfield while Salford won by the same score against Hartlepool thanks to an early goal and a late one from Tom Elliott and Matthew Willock, respectively.
Leyton Orient and Walsall both had to settle for a point in a 0-0 draw.