There was nearly a late contender for the goal of the season inside the first two minutes, when Kobbie Mainoo weaved and slalomed through a sea of Brentford bodies before finding Amad Diallo with the goal seemingly at his mercy.
It deserved a finish, but the Ivorian international was denied by a smart block on the line by Sepp van den Berg.
Shortly afterwards, it was Caoimhin Kelleher’s turn for heroics, with goal-line technology confirming that he had clawed away Harry Maguire’s header in the nick of time.
But it was an ominous start for Keith Andrews’ side.
Casemiro provided the breakthrough in the 11th minute, Maguire turning provider on this occasion as he nodded a Bruno Fernandes cross back across the box into the path of the Brazilian, who headed in his ninth PL goal of the season, further improving on what was already his best-ever goal-scoring campaign.
Another in-form scorer, Igor Thiago, had the chance to supply a rapid response, but somehow missed his kick at the far post under pressure from Luke Shaw.
Thiago had two more big chances before the half was over, and Senne Lammens also had to be very sharp to prevent an Ayden Heaven own goal.
But in an open contest, only the offside flag denied Amad at the other end, before Benjamin Sesko finally found the hosts’ second on the brink of half-time, courtesy of the latest Fernandes assist, curling past Kelleher after sitting a defender down.
The Portugal star continued to pull the strings in the second half, knowing that one more assist would move him level with the single-season PL record held by Thierry Henry and Kevin De Bruyne.
But it was the London outfit knocking hardest on the door, with Thiago miscueing yet another chance, relying on the offside flag to spare his blushes.
Dango Ouattara could then only find the woodwork from point-blank range, before Maguire bravely got in the way of his acrobatic attempted rebound.
The contest appeared to be drifting, but Mathias Jensen made a game of things when he arrowed a great strike into the corner from the edge of the box in the 87th minute for a career-high fifth goal against the Red Devils.
That set up a nervy finish for the hosts, but they held on for a valuable win.
United have now scored in 22 straight PL games, and face the tantalising prospect of potentially securing UCL qualification with a win against Liverpool at Old Trafford next weekend, if results elsewhere do not confirm it sooner.
Brentford’s winless run stretches back to February, although they remain just two points adrift of sixth, with a first-ever European qualification still on the cards.