Velicka's first goal for his new club gave a Rangers side weakened by injury and suspension a confidence-boosting three points to kick off their SPL campaign.
But full credit should go to goalkeeper Allan McGregor, who pulled off a priceless save from Michael Higdon's hesitant penalty with the score tied at 0-0.
"Obviously it was an important save for us," admitted manager Walter Smith afterwards.
"We didn't actually play all that well but managed to get the win which was the important thing especially after what happened during the week."
Rangers made a positive start with Smith discarding his usual caution by fielding an attacking line-up.
Fans' favourite Kris Boyd and Velicka formed a two-man strikeforce with summer signing Kyle Lafferty pushing up from midfield and there were no signs of a midweek hangover as Rangers dominated early on.
Home skipper Darren Barr had to make a desperate last-ditch tackle to deny Boyd as the striker prepared to pull the trigger and it took Falkirk 17 minutes to make any real impression.
Even then, Higdon failed to capitalise on a wild shot from Scott Arfield which fell kindly for him inside the penalty area.
Rangers ended the first period as they had started it and, after keeper Robert Olejnik pulled off an excellent save from Kevin Thomson's low left-foot drive, the home defence was required to scramble away an effort from veteran Christian Dailly.
Steven Whittaker's driving runs from the back were causing Falkirk problems, but the former Hibs man blotted his copybook in the 60th minute with a clumsy challenge on Kevin McBride.
Higdon failed to make Rangers pay and, even though the Ibrox side never looked totally convincing, there was a certain inevitability about their winning goal.
Boyd battled hard for possession and then fed Lafferty whose cut back was slotted home at the back post by Velicka.
Boyd's looping header almost killed Falkirk off and at the other end Kirk Broadfoot made a perfectly judged penalty box challenge on Higdon as time ran out.