The midfield player's 35th-minute goal proved sufficient against a lightweight Livingston side who had Paul Dalglish sent off late in the game. Livi player-manager Paul Lambert even picked himself in a bid to strengthen his midfield.
It was the home side who started in the more positive fashion but it was not until the 25th minute that there was a shot on target from either team.
Allan Ferguson reacted well to turn a Wes Hoolahan shot past his post and, moments later, Harald Pinxten looped a header a yard wide.
But Falkirk had the better chances after that. Daniel McBreen should have done better than lash a 16-yard shot over the top after a dummy from Pedro Moutinho.
McBreen returned the compliment by setting Moutinho in on goal but he fired the ball tamely into McKenzie's arms.
But Falkirk took the lead ten minutes from the break when Pinxten misplaced a header into the path of O'Donnell and he shot low into the right-hand corner of McKenzie's net.
Moutinho had another opportunity before the break from a long ball by Patrick Cregg but McKenzie's stood up well to save.
Livingston had a let-off four minutes into the second half when McKenzie saved brilliantly from a Craig Ireland header only for McBreen to send the rebound against the underside of the bar.
Russell Latapy should have scored after 58 minutes when O'Donnell set him up but he fired over the top.
Livingston's cause was not helped when substitute Dalglish was sent off 11 minutes from time for stamping on Cregg.