Following a lacklustre first-half display lacking any real quality from the hosts, centre-back Maguire broke the deadlock with a bullet header in the 72nd minute, before Onana saved a last-minute penalty from Jordan Larsson - the son of former United forward Henrik Larsson - to help the Red Devils secure a vital victory in dramatic fashion.
Erik ten Hag's side suffered back-to-back defeats in their opening two Group A games, but success over Copenhagen has restored belief in their quest to qualify for the knockout rounds and has helped them climb above their Danish counterparts into third place, just one point behind second-placed Galatasaray, who lost 3-1 to group leaders Bayern Munich earlier this evening.
Man United's victory took place in their first home fixture since the passing of club legend and footballing icon Sir Bobby Charlton at the age of 86 on Saturday.
Before kickoff, a lone piper lead a wreath-carrying Erik ten Hag, former goalkeeper Alex Stepney and academy captain Dan Gore towards the centre circle, before a perfectly observed silence was held to commemorate the career and life of Charlton.
It was Copenhagen who started the brighter of the two sides and the Danish champions were denied an early goal by the woodwork inside the first five minutes as Diogo Goncalves stuck an effort against the base of the post following a powerful driving run through United's defence from Mohamed Elyounoussi.
Copenhagen, who looked confident and comfortable in possession, were by far the better team in the first half, while the Red Devils were sloppy on the ball and struggling to adjust to the visitors' aggressive press.
The Red Devils lacked any form of invention in the final third, much to the frustration of the Old Trafford faithful, and even referee Marco Guida seemed to have had enough as he bizarrely blew the half-time whistle around 15 seconds before the 45-minute mark.
Danish playmaker Christian Eriksen replaced Sofyan Amrabat at half time in an attempt to provide a much-needed creative spark in midfield, and the Old Trafford crowd expected an immediate reaction from Man United, but it was Copenhagen who began the brightest once again.
Five minutes into the second half, Onana was called into action to make a superb one-handed save to prevent Lukas Lerager from finding the far corner with a fierce shot from the edge of the area.
Grabara then made an equally impressive one-handed shot at the other end just a few minutes later to prevent a low-driven strike from Eriksen creeping into the bottom corner.
More to follow.