Former Bairn Mark Roberts scored a penalty to knock Falkirk out of the Scottish Cup.
Darren Dods was penalised in the 57th minute for handball in the box, giving away Falkirk's third penalty in three games.
Roberts sent McGovern the wrong way with the spot-kick that will see Hibs head to Somerset Park next month.
The Edinburgh club's manager, Pat Fenlon, and assistant, Billy Brown, were both present at the match to scout out their potential opponents and saw Falkirk take the lead after six minutes through Blair Alston.
The midfielder had been lacklustre against Morton but he was vastly improved in this fifth round tie.
He diverted a drilled Tam Scobbie cross into the net with a looping header that floated into the top corner, catching Kevin Cuthbert off guard.
For the first half, Falkirk were the dominant side and should have had a penalty when Craig Sibbald was taken down by Eddie Malone in the box. Curiously, Crawford Allan handed Bairns a corner and Sibbald took his time in recovering from the blow.
It was one of a number of refereeing decisions that fell Bairns’ way - but, crucially, Dods' handball in the second half wasnt one of them.
By then Ayr had drawn level when, after McGovern had pushed a powerful Adam Dodd strike on to the crossbar, the ball bounced on the line and was nodded in by Andy Geggan.
Kieran Duffie was galavanting down the right throughout the first half but, on his return to the first team, Farid El Alagui couldn't convert and looked off the pace.
He did, however, test Cuthbert in both halves, as did Sibbald.
Willie Gibson was Bairns' main creator and linked up well with Mark Millar, Stewart Murdoch, Duffie and Sibbald.
But, when his lack of games kicked in midway through the second half, the Bairns toiled going forward.
After the break, Ayr burst into life and looked far from a team currently at the bottom end of the Irn-Bru First Division. An exciting cup-tie became a one-team show with the hosts pestering Falkirk and panicking Murray Wallace and Dods in the defence.
Their persistence paid when Dods' hand hit the ball. Falkirk claimed it was accidental but Allan pointed to the spot and Roberts converted.
United should have gone further ahead - twice - but were foiled on both occasions.
Geggan hit the post with a header at an empty goal after ghosting in to meet Adam Dodd's cross.
Then Michael McGovern was also relieved to hold the ball when another cross from Dodds looped high over him. The goalkeeper claimed the ball then seemed to stumble over the line, but ref Allan ruled it hand’t crossed.
Falkirk threw Jay Fulton, Rhys Bennett and Jordan White into the attack but couldn't find the net.
The best effort came from El Alagui's excellent header from a good Alston cross that Cuthbert pushed away with both hands.
But, as the intensity - and atmosphere - increased at Somerset Park, the Bairns couldn't capitalise, and both Wallace and Millar were fortunate to stay on during the added four minutes added after rash challenges.
For more see The Falkirk Herald next week.