United were dealt a blow when Johnny Russell was shown a straight red card after just half an hour at Tannadice on Saturday.
Despite the numerical disadvantage, the Tayside club did have chances to find the back of the net but it was Lafferty who netted the decisive goal to hand the Scottish champions all three points.
Rangers were without injured striker Nikica Jelavic but were boosted by the news that Dorin Goian, Lafferty and Steven Naismith were all fit to start despite concerns over their fitness.
New signing Matt McKay had to settle for a place on the bench following his arrival from Australia earlier in the week.
United were without Sean Dillon through injury for the clash at Tannadice, while the visit of the Ibrox side came too soon for Jon Daly, who remained on the sidelines.
The home side could have taken the lead early on when Willo Flood's shot took a deflection off Steven Davis and fell just wide of Allan McGregor's left-hand post.
Garry Kenneth picked up the first booking with just nine minutes on the clock for a blatant push on Kirk Broadfoot, as Rangers gradually began to impose themselves on the game.
Juan Manuel Ortiz set up Lafferty but his shot lacked power and was gathered by goalkeeper Dusan Pernis, before Naismith blasted over from close range from a Gregg Wylde cross.
At the other end, McGregor had to react quickly to block an effort from distance from Danny Swanson, who was booked moments later for diving after going to ground under pressure from Goian.
It was proving to be a busy afternoon for referee Callum Murray and Russell was shown a straight red card for a head-butt on Broadfoot, who was also cautioned for his part in the altercation.
Rangers had a decent chance to open the scoring when Naismith raced onto a Maurice Edu pass only to lash inches wide of the target with a well-struck effort, before Lafferty was denied at the near post by Pernis.
Rangers emerged for the second half determined to put their hosts under pressure and Lafferty was not too far away with a shot that was blocked by Kenneth following an inviting ball into the box by Davis.
The Northern Irishman then squandered another chance when he met a Naismith cross only to send a header flashing across the face of goal and past the upright.
Lafferty finally hit the target with 60 minutes gone after collecting a decent ball from Wylde and nodding past the grasp of Pernis from 10 yards.
Only a great save from the goalkeeper prevented a Davis header from finding the back of the net, before the midfielder surged into the box and saw his shot smack off one post before crashing off the other as he was again denied adding his name to the scoresheet.
Rangers continued to push for a second goal and Kenneth came to the rescue of United when he hooked a goal-bound effort from Naismith to safety.
The Tannadice side also looked capable of finding the back of the net and a long-ranger from John Rankin skimmed the outside of the post.
Gavin Gunning then nodded over in the final seconds of the game but Rangers were able to hold on to ensure they returned to Ibrox with maximum points.