Having replaced the injured Phil Jones, substitute Smalling scored with his first touch thanks to a sixth-minute header, but the visitors dug deep to make light of the gulf in stature between two clubs scrapping at opposite ends of the Premier League.
Danny Ings responded with his fifth goal in his last six league games to equalise and went close to doubling his tally as second-bottom Burnley threatened to cause a major upset in their survival fight.
Smalling, though, eased the nerves of the home faithful just before the interval with another header and Robin van Persie converted a late penalty to propel United up to third.
Defeat was another harsh Premier League lesson for Burnley – winless at Old Trafford since 1962 – although the character on show from the Lancashire club bodes well for the relegation run-in.
Paddy McNair, fresh from this week signing a new deal, and Jonny Evans came in for United at the expense of the suspended Luke Shaw and Antonio Valencia.
Burnley, forced to change their starting XI for the first time in 11 games as Dean Marney's season-ending injury paved the way for Michael Kightly to deputise, went close early on as former United defender Michael Keane headed Kieran Trippier's third-minute corner wide.
And United punished Sean Dyche's men three minutes later as Smalling made an instant impact from the bench to break the deadlock.
Tom Heaton, another Burnley player with Old Trafford connections, palmed over McNair's 35-yard volley to force a corner that Wayne Rooney whipped in. Headed flicks from Van Persie first at the front post and then Radamel Falcao at the back post eventually dropped kindly for Smalling and he nodded across Heaton and into the net.
While Dyche will have been disappointed to see his side concede yet again from a set-piece, Burnley's response was typical of the spirit he has instilled into his players.
Kightly fired tamely at David de Gea before Ings continued his recent purple patch in front of goal to make it 1-1 with 12 minutes gone.
Trippier's superb cross from the right looped over Smalling and Ings was lurking at the back post to steer a neat header past De Gea.
Ings then had a goalbound effort cleared following another Trippier corner and Ashley Barnes curled just wide from 20 yards as Burnley enjoyed a dominant spell.
United were dealt a second injury blow of the first half when Daley Blind departed six minutes before half-time, replaced by Ander Herrera.
For all Burnley's supremacy, however, familiar defensive frailties returned to haunt them on the stroke of half-time as Smalling edged United back in front.
Rooney and Angel di Maria played a short corner and the latter stood up a cross to the back post that Smalling met with an unstoppable header.
The second half failed to match the levels of a breathless first period, although that could have been very different had Ings not shot straight at De Gea from close range immediately after the restart as George Boyd's cross forced a defensive mix-up between Smalling and Herrera.
It was left to Van Persie to put the result beyond any doubt with eight minutes remaining after Di Maria cut in from the left and was clumsily upended by Scott Arfield, the Dutchman sending Heaton the wrong way from the resulting penalty.
Heaton then kept out a rasping Di Maria drive and Adnan Januzaj's follow-up was well blocked by Jason Shackell as United well on top.
