The visitors finished the game without skipper Darren Barr after their skipper was red-carded following two second-half bookings and United's victory should have been more comfortable as they passed up a string of chances to net a second.
However, the Bairns still came within a whisker of snatching an unlikely draw when Scott Arfield struck the post 16 minutes from time.
United manager Craig Levein made two changes from the side which eked out a 0-0 draw with third-place rivals Hearts last weekend, drafting in striker Warren Feeney and full-back Sean Dillon in a more attacking formation.
Falkirk manager John Hughes, who in the build-up to the game reiterated his desire to work in England in future, also made two alterations bringing back experienced full-back Jackie McNamara and striker Michael Higdon.
However, despite a determined opening, only the linesman's raised flag saved Falkirk from going behind in only the fifth minute.
Paul Dixon's left-wing cross was flicked goalwards by Feeney but when his effort rebounded and Jon Daly bundled it in from a yard out the 'goal' was disallowed for offside.
When Daly's deft touch darted across the goalmouth from another Dixon cross six minutes later, it was clear Falkirk would have to do something to nullify the threat of United's potent left flank.
And United's pressure eventually paid off with the breakthrough in the 22nd minute.
A magnificent pass from Dixon sent Feeney clear and when Barr's clearing header only made it to the edge of the area Robertson followed in to side-foot his volley past Robert Olejnik.
The Austrian keeper was again put under immense pressure from Craig Conway's in-swinging corner in the 25th minute but the crossbar saved the Bairns twice in an aerial scramble.
There was a glimmer of hope for Falkirk in the 29th minute as Arfield released Steve Lovell through on goal but United keeper Lukasz Zaluska did enough to force the striker away from goal and the chance was gone.
The opportunity appeared to give Hughes' struggling outfit a lift as they finally strung together some neat passing movements and put pressure on the home goal, without really carving out any clear-cut openings.
However, they finally created an opportunity two minutes later, only for Arfield to send Michael Higdon's pass wildly over from an angle.
Another dangerous Conway corner at the other end caused Falkirk major difficulties and they had Arnau Riera to thank for his goal-line block on Garry Kenneth after Daly's header.
The Bairns knew they needed a massive improvement from their first-half display and they almost snatched an immediate equaliser.
Arfield played a neat one-two with Lovell on the edge of the box but Zaluska, playing for the first time since signing a pre-contract agreement with Celtic, stopped the midfielder's low shot with his fingertips.
United were aware another goal would kill the game and they came close on three occasions in the space of only 11 minutes.
First, Conway struck the junction of the bar and post with a mis-hit half-volley in the 51st minute and then they should really have sealed all three points six minutes later as Feeney scampered on to Willo Flood's pass, only for Daly to blast his cutback too high.
When Feeney headed Conway's right-wing cross just over it followed Higdon's drive wide of the target and the striker's net-bound header which was held by Zaluska as Falkirk enjoyed some meaningful possession.
Falkirk then survived a penalty claim in the 69th minute as Barr appeared to bundle over Feeney.
Arfield had proved himself Falkirk's most dangerous player throughout and the 20-year-old came within a whisker of an equaliser with 16 minutes remaining.
Higdon's flick-on was headed into Arfield's path by Lovell but the midfielder's volley with the outside of his right foot swerved just too much and rebounded off Zaluska's left-hand post and across the goal to safety.
Falkirk's chances of staging a late comeback appeared to be dashed with seven minutes remaining when skipper Barr was sent off.
The Scotland defender, who had been cautioned for a foul on Daly in the 67th minute, received his second yellow card when he chopped down Flood on the half-way line.
Falkirk still piled men forward in search of an equaliser but, despite some unorthodox defending at times, United held on for all three points.