On this occasion it was Wenger's successor Unai Emery who would had more to be satisfied as his Arsenal side stretched their unbeaten run to 20 matches - although United boss Mourinho also took comfort from a result that left his side 18 points behind leaders Manchester City.
Arsenal has not won a Premier League match at United since 2007, and drops fifth with 31 points.
The Gunners went ahead when keeper David de Gea made a awful mistake - albeit on a greasy surface - after he could only parry Shkodran Mustafi's 26th-minute downward header that Ander Herrera could not clear before it crossed the line.
Mourinho has been tasked with leading Manchester United to a trophy this season after they ended up second and without any silverware at all last term.
However, it took United just four minutes to find an equaliser.
The home side were rewarded with an equaliser just four minutes later, as Anthony Martial slotted home a cut-back from Herrera which should have been cleared before it reached the Frenchman.More news: XFL style, starting in 2020
Arsenal nearly added a third goal in a matter of minutes, with Pierre-Emerick Aubameyang being superbly denied by De Gea from close range before Mkhitaryan somehow fired over from no more than three yards out. Rob Holding, we are going to wait for the doctor but maybe it's a big injury'.
'When we are winning it was a good moment but they equalised very quickly then we got up after their two goals.
James Milner levelled for the Reds after Jack Cork's surprise opener and substitute Roberto Firmino set Liverpool on course for a 12th win in 15 league games with his first touch before Xherdan Shaqiri rounded things off. It's his knee, ' Emery said after the game.
However - like in the first half - United found an immediate equaliser.
Arsenal, meanwhile, have made two changes from the team which started the win over Tottenham on Sunday. Arsenal failed to deal with a long ball from the kickoff and after Kolasinac diverted the ball towards his own goal, Lingard stole in to poke a shot past Leno.
"We scored four goals and we drew 2-2, but we score four goals and even in matches like today where we play well we always shoot ourselves", Mourinho told a news conference.