Martial created an opening with some neat footwork, with his shot blocked by a sliding Grandison.
United’s pressure finally paid off with half-time approaching courtesy of Smalling.
After Borthwick-Jackson had powered down the left, his cleared cross was headed back in by Morgan Schneiderlin and Smalling’s shot on the turn deflected in off Nathaniel Knight-Percival.
As Shrewsbury attempted to steady the ship, they conceded the second goal on the stroke of half-time.
Zak Whitbread had tripped Martial and Mata curled the resulting free-kick over the Shrewsbury wall and into the net with Leutwiler rooted to the spot.
Early in the second half, Lingard produced a good touch to control a cross as he looked to extend United’s lead but Brown made a another important clearance for Shrewsbury.
United were still making all the running and Depay made another bid to get on the scoresheet when he cut inside and let fly with a shot that deflected behind for a corner.
Van Gaal’s men grabbed their third goal on 61 minutes when they counter-attacked with pace from a rare Shrewsbury break.
Mata’s touch sent Ander Herrera away and his cross from the left was turned home by Lingard, who produced a smart, first-time finish.
Lingard missed a chance to add to his tally when he bundled a close-range shot wide as United continued to press.
And substitute Will Keane then saw a decent snap-shot come back off the post.
United ended the game with 10 men when Keane was forced off with an injury after all three substitutes had been used.