The Bairns, who came into the match having taken the scalps of Hearts, Inverness Caley Thistle and Motherwell in recent times, were quickly into their stride and a prolonged period of possession created space for Scott Arfield, but the teenager hit his shot poorly making it easy for keeper Alan Combe.
The Falkirk opener was only delayed by three minutes after more good passing allowed Dean Holden time to measure his cross from the left for Finnegan to glance the ball past Combe at the near post.
Killie were poor, making mistakes at both ends of the pitch, and it cost them the chance of an equaliser in the 17th minute after a slip by Darren Barr left Tam Scobbie all alone to face the charging Gary Wales and Colin Nish, but Wales' attempted pass hit the defender and went out of play.
Killie's best piece of play created a clear chance when a good ball from Fraser Wright found David Fernandez in space in the box, but in keeping with Killie's display, he shot well over under no pressure.
It was another Killie mistake that gave Falkirk a chance to double their lead when Wright gave the ball away deep in his own half, but when the cross came over, Michael Higdon's header was badly timed and flew over.
Home manager Jim Jefferies sent on new signing Alan Morgan for the ineffectual Fernandez at half-time, and it was the winger's set-pieces that looked to provide Kilmarnock with a route back into the game.
His fizzed corner was met at point-blank range by Killie's best player Jamie Hamill but he volleyed over.
Falkirk's introduction of Russell Latapy seemed to have secured their superiority, but they faced a late flurry as the home side tried to rescue a point.
A Wales cross found Wright in a great position at the far post, but his downward header was a yard wide.
The Killie fans' frustrations intensified when Simon Ford shot home only to see the offside flag raised against striker Nish.
Killie's desperation resulted in a goalmouth fracas in the final minute when Wright charged in late on goalkeeper Tim Krul, who reacted badly and both players were shown the red card.