Vincent scored a goal in either half to give Martin Allen's side a narrow victory over Sammy McIlroy's League Two Morecambe.
Cheltenham started much brighter than they did on Saturday and pressed forward in the early stages.
But they were caught cold in the fifth minute when Danny Carlton was fouled 30 yards out. Craig Stanley hit a long free-kick to the far side of the box, from where Henry McStay rose to loop a header back over Scott Brown to give the Shrimps the lead.
Cheltenham worked hard to get back into the game and Barry Hayles once again proved what a valuable signing he could be for the Robins as he inspired a 22nd minute equaliser.
He used all his strength and guile to hold off Dave Artell before hitting a shot that Barry Roche did well to parry, but the ball fell perfectly for Vincent to fire in the equaliser.
But Morecambe were back in the lead after 27 minutes when top scorer Rene Howe forced his way into the box only to be brought down by Darren Kenton for a penalty, which he converted himself.
The home fans were irate ten minutes later though when referee, Mr Foster, stunned everyone by awarding a controversial Cheltenham penalty after a melee in the Morecambe box.
Nobody appealed but he signalled there had been a push and awarded the spot-kick that was converted coolly by visiting skipper John Finnigan to make it 2-2.
The home side hit back again with Carlton and Howe wasting glorious chances to give their side the lead again and they were made to pay as the Robins went ahead for the first time in the game on 54 minutes.
Ian Westlake was given too much room on the left and he hit a great deep cross which was met at the far post by Vincent, who stretched to poke home his second of the night.
Morecambe poured forward looking for an equaliser and Carlton, Howe and Michael Carr all forced Brown into superb saves to keep his side's advantage and earn them a third-round spot.