There were only eight minutes gone when Accies defender Chris Swailes, under no pressure, carelessly knocked the ball past his own goalkeeper Tomas Cerny to give the home side the lead.
It was the worst possible start for Billy Reid's side but they responded positively and Simon Mensing volleyed in the equaliser before the break.
Accies looked the more likely after the break and 15 minutes from the end the comeback was complete when McGowan, on loan from Celtic, stretched to knock in a pinpoint James McCarthy pass to leave Falkirk only one point ahead of the Highlanders - who beat Hibernian.
The Bairns will know they face a battle for Scottish Premier League survival - but they were handed a great start today.
In the eighth minute, seconds after Falkirk goalkeeper Dani Mallo had made a fine save from New Douglas Park striker Richard Offiong, Swailes knocked the ball back to Cerny without looking.
The defender watched in horror as the ball trundled past his goalkeeper, who was on his way out to gather.
It was a basic mistake by the visitors but they responded positively, drawing level from a Brian Easton corner in the 18th minute after McCarthy had hassled Bairns defender Steven Pressley to concede the set-piece on the byline.
Mallo flapped at Easton's inswinging cross inside a busy six-yard box and when the ball dropped to Mensing 10 yards out, the Accies midfielder swiftly volleyed it back high into the net.
In the 29th minute of a relatively open but unexciting game, Bairns midfielder Scott Arfield found himself with the ball at his feet in the Hamilton box but he failed to connect properly with his left-footed drive and Cerny saved comfortably down at his left-hand side.
Seven minutes from the interval Offiong woke the game from its slumber when he turned 30 yards from goal and volleyed just wide of the target.
And a minute before the break McCarthy broke past Arfield in the Falkirk box but his cross was knocked past by Offiong under pressure from Bairns skipper Darren Barr.
Falkirk striker Michael Higdon, who had been anonymous in the first half, was replaced by midfielder Patrick Cregg at the interval with McCann moving up to join Steve Lovell in attack.
There was no immediate improvement in the Falkirk play and five minutes after the restart, as Accies probed at the edge of the home side's penalty area, Mensing fired in a side-footed drive which Mallo saved with ease.
Arfield was booked for a foul on McCarthy on the left-hand side of the Falkirk box in the 53rd minute but from Easton's delivery to the back post, Swailes headed high over the bar.
Falkirk broke seconds later when midfielder Carl Finnigan sent Lovell clear though on goal with a raking pass.
The former Dundee striker rounded Cerny but his touch was too strong and he could only strike the post with his angled-shot before Burton O'Brien took the pressure of the visitors when he blasted over the bar.
Accies defender Mark McLaughlin, who crashed in to the woodwork as he attempted to clear, came back on after treatment and moments later watched McGowan volley over from 12 yards from Mensing's cross. Accies looked the stronger side and they stretched in to the lead in the 75th minute after McCarthy played the pass of the match over Barr to find McGowan who stretched to poke the ball past Mallo from 10 yards out.
Moment later, with Accies going all out to clinch the three points, midfielder James McArthur drove the ball in from the edge of the box and it clipped Finnigan before hitting the post.
As Falkirk rallied, Cerny pulled off a fine save from Cregg before McLaughlin was stretchered off with a head knock, to be replaced by James Gibson, after clashing heads with Bairns substitute Mark Stewart.
But there was no real conviction about Falkirk in the seven minutes added on and Accies held on with a degree of comfort.