Opponents St Mirren survived an early onslaught and had several chances but sub Roy O'Donovan lofted the ball over Mark Howard with five minutes left.
Craig Conway soon lashed home a second as the United fans hailed their latest Scottish Premier League win by singing Thompson's name.
A bumper 11,378 crowd turned up to pay tribute to Thompson, who died following a long battle with cancer, after admission prices were cut to £5.
Thompson's favourite music, including the Beatles and Rod Stewart, was played before kick-off along with Olivia Newton-John's 'Hopelessly Devoted', which summed up the late chairman's attitude to his club.
After a minute's silence, United came out determined to secure a win in Thompson's memory.
Conway put in a dangerous ball just in front of Jon Daly in the opening seconds and Darren Dods had four efforts on goal from seven corners inside eight minutes.
The defender's best effort, a volley that followed a Scott Robertson flick-on, forced Mark Howard to make a fine diving save, and Dods headed wide from close range seconds later.
Conway's deliveries were causing havoc and the winger saw Howard tip over his powerful near-post strike after a half-cleared corner.
United had started with the confidence of team on the back of four victories but St Mirren were looking for a third consecutive win and soon took the sting out the game.
Manager Gus MacPherson had brought in defender Scott Cuthbert as he returned to the 3-5-2 formation that yielded victory over Rangers, and Saints kept possession for long spells.
They almost took the lead in the 24th minute when Hugh Murray caught Dennis Wyness' high and deep cross on the volley 14 yards out. Lukasz Zaluska got down well to parry and Wyness slid in to poke wide as Lee Wilkie dallied on the ball six yards out.
St Mirren continued to hold their own in a scrappy passage of play amid blustery conditions, although United striker Francisco Sandaza shot just over from 20 yards in the final minute of the first half.
The second half was much more stretched as United pushed more men forward.
Daly had a good penalty claim denied when he appeared to be pushed by John Potter.
Saints immediately broke to create their best chance. Billy Mehmet escaped his marker to run onto Murray's lofted ball down the right but Zaluska came off his line to block the striker's shot.
The visitors broke with numbers again on the hour mark and Franco Miranda strode clear down the left following smart play by Garry Brady and Mehmet. But the Argentinian was caught between shooting and crossing and dragged his effort wide of the far post.
Dods went close three minutes later after another Conway corner when his header looped on top of the bar, shortly before he left the field for treatment to a head wound.
Craig Levein brought on Garry Kenneth, while Roy O'Donovan and Danny Swanson replaced Daly and Willo Flood.
Chances continued to fall at both ends and Scott Robertson sliced a shot inches wide of Howard's post following a long throw. Morgaro Gomis then cleared off the line after Murray met Brady's corner on the volley 18 yards out.
An unmarked Gomis soon volleyed Conway's corner towards goal from similar range, but Howard comfortably clutched the ball.
Wilkie blocked Craig Dargo's shot after the St Mirren substitute had worked space. The same player headed just wide at the other end after winning Paul Dixon's flighted free-kick inside the six-yard box.
The breakthrough eventually came in the 85th minute when on-loan Sunderland forward O'Donovan grabbed his first goal for United.
The Irishman's perseverance allowed him to wriggle inside two defenders on the byline and chip the ball over Howard's head and into the far corner via a deflection.
Conway wrapped up the win in the last minute on the counter attack, ignoring team-mates on either side of him to fire into the corner from the edge of the area.
Zaluska ensured his fourth clean sheet in five games in injury-time by pushing clear substitute Stephen McGinn's long-range strike.