The Red Devils welcomed the Welsh side to Old Trafford knowing that a win would solidify their position in the top four, especially given that two of the teams directly below them would be squaring off in a London derby on Sunday.
Swansea, meanwhile, are fighting for Premier League survival.
United kept the ball in the Swans' half for the opening five and their early dominance would pay off when Romelu Lukaku was found in the box by Alexis Sanchez. The Belgian wasted no time hitting it past the Swansea stopper for his 100th Premier League goal.
Lukaku also joined an exclusive club with his opener, becoming just the 28th player to score 100 in the English top flight.
The home side doubled the lead in the 20th minute following a quick breakaway, with Jesse Lingard finding Sanchez with a great through ball, putting him through on goal to score what was only his second since his move from Arsenal in January.
An opportunity for a third would fall to United 10 minutes later, but Lingard would shoot wide after creating space with some nifty footwork at the top of the box. Jose Mourinho's men headed into the half with a two-goal advantage, while the Swans were left to plot a seemingly impossible comeback.
Swansea came out the stronger-looking side in the second period, but David de Gea made two incredible saves from two Tammy Abraham shots to keep the lead intact.
Lukasz Fabianski came up with a huge save of his own after Paul Pogba picked Lukaku out in the box with a pinpoint aerial ball in the 81st minute. The forward got off a shot from close range, but the former Arsenal keeper got the better of him this time around.
Mourinho's men would hold on to their lead until the final whistle, leapfrogging Liverpool to reclaim the second spot on the table after the Reds' earlier win against Crystal Palace had left them above their bitter rivals.