Kilmarnock came back from a goal down to clinch victory in a pulsating encounter at Rugby Park.
Anthony Stokes fired home the opening goal but his side lost
Anthony Stokes fired the visitors in front in 42 minutes after his deflected shot beat Graeme Smith.
But Garry Hay equalised for Kilmarnock in 59 minutes with a fine strike from 12 yards into the top left-hand corner.
Kilmarnock took the lead in 70 minutes when Stephen O'Donnell headed into his own net. Nish was then sent off after receiving a second yellow for diving.
Kilmarnock midfielder Allan Johnston returned to the starting line-up, after the former Sunderland man missed the defeat at Dundee United last week with a slight knock.
Falkirk defender Jack Ross, who limped out of the draw with Hearts on Monday, was fit enough to take his place in the team.
The game started in lively fashion, with Colin Nish almost running clear of the Falkirk defence in the opening minute.
A quick Peter Leven free-kick almost caught out the visitors' defence and then Jeroen Lambers caused panic when he dropped a Steven Naismith header.
But Lambers made up for his mistake by pulling off a good save from Danny Invincible testing header.
Johnston was robbed of Kilmarnock's opener in 10 minutes when Lambers produced another good stop to keep out his header.
Colin Nish received a second yellow card after tumbling in the box
But it was the visitors who almost opened the scoring three minutes later, with Steven Thomson clipping the top of the crossbar after an Anthony Stokes cross.
Stokes fired the visitors in front three minutes before the interval, when his 20-yard shot was deflected off Killie defender Gordon Greer to send Smith in the wrong direction.
Falkirk looked dangerous on the break with Latapy pulling the strings in midfield.
However, Kilmarnock got back on level terms when Leven's cross was only partially cleared by the Falkirk defence.
Hay ran in unchallenged and scored with a rasping drive from the edge of the box.
Kilmarnock then took the lead in 72 minutes, courtesy of an own-goal from O'Donnell.
The midfielder got his head to a Leven free-kick as the ball flew into the net.
Latapy had a chance to equalise when he took a free-kick 22 yards out, but hit his shot against the Kilmarnock wall.
The game ended in dramatic fashion when Nish was sent off after receiving a second yellow card for diving in the box.
Kilmarnock: Smith, Grant Murray, Hay, Frazer Wright, Greer, Fowler, Johnston, Leven, Invincibile, Naismith, Nish. Subs Not Used: Combe, Ford, Dodds, Di Giacomo, Stephen Murray, O'Leary, Adams.
Sent Off: Nish (88).
Booked: Greer, Nish.
Goals: Hay 59, O'Donnell 70 og.
Falkirk: Lambers, Ross, Twaddle, Milne, Barr, Cregg, Thomson, O'Donnell (John Stewart 75), Latapy, Gow, Stokes. Subs Not Used: Higgins, Dodd, Craig, Lima, Scobbie, McManus.
Booked: O'Donnell, Thomson, Cregg, Barr, John Stewart.
Goals: Stokes 42.
Ref: C Richmond