The Clarets had managed to shut out United for the entirety of the first half, but eventually fell behind in the 54th minute as Anthony Martial lashed in a vicious strike.
Thirteen minutes later, Phil Bardsley was booked for a poor tackle on Smalling, and the centre-back reveals it became the plan to focus on the former Red Devil (via the club's website).
"I think as soon as Bardo [Bardsley] was shown a yellow card, the instructions were just to give it to Anthony [Martial] as soon as we can because when he is running at people, he is a danger. His goal was brilliant.
"I think him and Marcus [Rashford] have both played on that left side and both have chipped in with goals and assists, and it's great for us that we have such great attacking flair."
The fearsome strike was Martial's ninth Premier League goal of the season and the Frenchman needs just two more to match his best record during his debut season.
Burnley displayed an admirable attitude and rallied after going behind, showing good attacking nous, but the visitors held on for all three points to make it four wins on the bounce.
"It was a battle and I think you know what you are going to get when you come here," Smalling added. "You are going to battle first before you can play. It was a great win because many teams come here and lose or draw.
"When the game is 1-0, it is in the balance and they had a couple of corners at the end, so it was a bit nervy but we did well as a team."
As well as United, only West Brom, Arsenal, Spurs and Liverpool have come away from Turf Moor with all three points.