It was United's biggest opening-day win since beating Fulham 5-1 at Old Trafford in 2006 and Mourinho's biggest league win as Red Devils manager.
Mourinho, though, insists he is not getting carried away. A few minutes later Pogba picked out Juan Mata with a glorious chip, though the Spaniard opted to pass rather than shoot, and allowed Angelo Ogbonna to clear the ball to safety.
There was no denying the Belgium striker six minutes later, though.
And the Argentinian paid a heavy price for that foul for, when Mkhitaryan floated over the consequent free-kick, that man Lukaku ghosted in front of Masuaku to nod home his second goal of the afternoon.
Earlier Tottenham capitalised on Jonjo Shelvey's senseless sending-off to beat Newcastle 2-0 in their season opener with Dele Alli and Ben Davies scoring in the final half-hour.
The 23-year-old became Mourinho's first signing back in June, signing for £31m from Benfica, and has featured as centre back in pre-season and against Real Madrid.
Rashford then hit the post from distance and Daley Blind saw a shot bounce over.
A third appeared inevitable and it arrived when Mkhitaryan slipped Martial in to score in the 87th minute as West Ham fell apart late on.
Lukaku scored twice on his league debut for the club following his move from Everton, while Nemanja Matic, a £35m buy from Chelsea, ran the midfield in a manner that made it hard to believe he spent most of the summer training alone at Stamford Bridge.
The game got off to a lively start, the ball bouncing off both teams as technique took a backseat to the classic Premier League swashbuckle.
A lot of the responsibility will fall on Romelu Lukaku, and the Belgian must have been delighted to see West Ham first up.
Jose Mourinho has said that new Manchester United signing Victor Lindelof needs time to adapt to the rigours of English football before he can regularly start in the Premier League.
Unfortunately for manager Slaven Bilic, West Ham won't be at full strength for this game. This was the kind of match United would have drawn last season, this season they ran riot.