United, in their first league game under new manager Jose Mourinho, followed last weekend’s Community Shield success against the champions Leicester City by winning 3-1 at Bournemouth overnight, with Zlatan Ibrahimovic scoring again.
Liverpool, looking likely at one point to be even more comfortable winners at Arsenal, eventually came through 4-3.
The old rivals, with 23 championships between them in the past 40 years, both missed out on cherished top-four finishes in May but have been widely tipped to improve on the back of heavy investment in the close-season.
United did not miss their world-record signing Paul Pogba, who was suspended, and Ibrahimovic linked well with Wayne Rooney.
Juan Mata, who some observers thought Mourinho would sell for a second time after letting him leave Chelsea, put United ahead in the first half before Rooney and Ibrahimovic effectively made the game safe.
Adam Smith pulled one goal back for Bournemouth.
“Very composed, very organised,” was Mourinho’s verdict. He must now decide whether to bring Pogba into the team for Friday’s televised home game with Southampton.
Liverpool, facing three away games to start the season because of building work at Anfield, could have been two goals down at halftime against last season’s runners-up Arsenal.
Goalkeeper Simon Mignolet saved a penalty by Theo Walcott, who then beat him two minutes later with the opening goal.
When Phillippe Coutinho equalised just before the interval, however, it proved to be the first of four goals by the visitors in an astonishing 17-minute spell.
Arsenal, missing central defenders Per Mertesacker, Laurent Koscielny and Gabriel, fell apart as Adam Lallana, Coutinho again and debutant Sadio Mane established what seemed an unassailable 4-1 lead.
There were brief doubts as Alex Oxlade-Chamberlain and Calum Chambers reduced the deficit but Liverpool held out without further alarm for only their second win in 21 visits to the London club.
“We can always score goals and we can defend much better,” said their manager Juergen Klopp.
“It’s the Premier League and nothing should be easy in this league. Nothing is.”
Manager Arsene Wenger, booed by supporters at the final whistle, said he was “working very hard to get one or two players in” before the transfer window closes on August 31.
With Leicester beaten on Saturday, statistical analysts Opta said it was the first time since 1953 that the English champions and runners-up had both lost their opening game of the season.
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