Frenchman Mousset fired the Blades ahead in the 51st minute and added a second two minutes later.
When Ben Osborn made it 3-0 in the 72nd minute there looked to be no way back for the home side.
But Devante Cole pulled a goal back in the 78th minute and fellow substitute Leya Iseka grabbed a second four minutes later to leave the Blades sweating before closing out a win that lifts them to 14th in the Championship.
In a scrappy first half, there was little action in the final third at either end.
Mousset looked the player most likely to score, dragging an early shot wide and also shooting into the side netting.
In the latter stages of the half, Mousset was the first player to put an effort on target, firing straight at Brad Collins from inside the area after meeting a low cross from Iliman Ndiaye.
Barnsley responded with an effort which went close as Victor Adeboyejo’s cross picked out Jasper Moon, whose header narrowly missed the target.
Callum Styles then found Adeboyejo who steered the ball goalwards at the near post, but Robin Olsen was able to gather it comfortably.
Cauley Woodrow shot straight at Olsen soon after the re-start.
At the other end, George Baldock saw his deflected effort land on the top of the net.
Mousset opened the scoring with a fine finish from the edge of the area after cutting inside and holding off a couple of defenders.
Mousset struck again two minutes later, firing into the roof of the net from close range after meeting a low cross from Osborn, who was allowed space down the left-hand side.
Barnsley boss Markus Schopp made a substitution, sending on Iseka in place of Dominik Frieser.
Another change for the hosts just before the hour-mark saw Cole take over from Adeboyejo.
United boss Slavisa Jokanovic made his first change as Mousset then made way for Billy Sharp.
Woodrow put a free-kick just over and Moon hit a post as the home side looked to try and force their way back into the game.
After an off-balance Cole screwed a shot wide, the visitors extended their lead.
The third goal came when Baldock sent over a low cross from the right and Osborn (72) had time to control the ball before firing beyond Collins.
Cole (78) pulled a goal back with a thumping shot to give Barnsley hope.
They received a further boost four minutes later when Iseka forced his way through in the area before firing past Olsen.
Moon put a header wide in stoppage time as United held on to claim all three points and leave Barnsley with just one win so far this season.
Register for free to continue reading
Registration is a free and easy way to support our truly independent journalism
By registering, you will also enjoy limited access to Premium articles, exclusive newsletters, commenting, and virtual events with our leading journalists
Already have an account? sign in