Porto started well, but it was Man United that made the breakthrough in the seventh minute, with Marcus Rashford's strike finding a route under the home side's goalkeeper Diogo Costa, who might feel that he should have done better.
Wonderful footwork saw Rashford, who now has four goals for the season, break into a dangerous area, and he found the back of the net to give the away side a huge lead in Portugal.
Rashford then turned provider for Rasmus Hojlund, as the Denmark international also squeezed his effort past Costa, with the striker off the mark for the new campaign to double Man United's advantage.
Porto, though, had one back in the 27th minute through Pepe, as the forward headed into the back of the net from close range after reacting to a smart save from Red Devils goalkeeper Andre Onana.
The hosts then levelled the scores in the 34th minute with a Samu Omorodion header from a Joao Mario cross, with Man United simply unable to deal with balls into the penalty box.
Omorodion was at it again in the 50th minute, smashing a Pepe cross into the back of the net to complete the turnaround, leaving the visitors stunned once again.
Bruno Fernandes was handed a second yellow card in the 81st minute for a high boot, with the Red Devils captain given his marching orders for the second match in a row.
Man United were pressing for a leveller, though, and it arrived in the 91st minute when Maguire headed a corner from Christian Eriksen into the bottom corner to potentially save Erik ten Hag's job.
PORTO VS. MAN UNITED HIGHLIGHTS
Marcus Rashford goal vs. Porto (7th min, Porto 0-1 Man United)
Man United make the breakthrough in the seventh minute of the contest, as Rashford races into the Porto box before firing into the back of the net, but there will be questions asked about the goalkeeper Costa, as it went through him too easily!
Rasmus Hojlund goal vs. Porto (20th min, Porto 0-2 Man United)
Man United double their advantage in the 20th minute of the contest, and Hojlund is off the mark for the season, with the Denmark international firing into the back of the net after being released by Rashford.
Again, there have to be question marks surrounding the goalkeeper, but Hojlund will not mind, and the visitors have taken a 2-0 lead inside 20 minutes in Portugal.
Pepe goal vs. Man United (28th min, Porto 1-2 Man United)
Porto are back in the match in the 28th minute, as Pepe heads home from close range.
Man United goalkeeper Onana did well to keep out a header from Omorodion, but Pepe was first on the scene to head into the back of the net. What an eventful first period!
Samu Omorodion goal vs. Man United (34th min, Porto 2-2 Man United)
Porto level the scores in the 34th minute of the match, as the dangerous Omorodion heads a cross from Joao Mario into the back of the net from close range; it is an excellent header from the forward.
Samu Omorodion goal vs. Man United (50th min, Porto 3-2 Man United)
Porto turn the game around early in the second period, as Omorodion sweeps a cross from Pepe into the back of the net. Again, it is terrible defending from Man United, who have simply fallen apart in this match.
Bruno Fernandes red card vs. Porto
Man United are reduced to 10 men, as Fernandes, who was sent off against Tottenham Hotspur at the weekend, picks up a second yellow card for a high boot.
Harry Maguire goal vs. Porto (91st min, Porto 3-3 Man United)
Incredible scenes here, as Man United level the scores in the 91st minute, with Maguire heading a corner from Eriksen into the bottom corner. The Red Devils have surely secured a late point here!
MAN OF THE MATCH - SAMU OMORODION
Omorodion proved to be too much for Man United in Portugal, with the striker scoring twice, but he could have had a couple more, with the visitors struggling to deal with him.
Porto won an intense race for the 20-year-old's signature over the summer, and he looks capable of becoming one of the best strikers in world football in the coming years.
PORTO VS. MAN UNITED MATCH STATS
Possession: Porto 46%-54% Man United
Shots: Porto 15-29 Man United
Shots on target: Porto 7-9 Man United
Corners: Porto 5-10 Man United
Fouls: Porto 7-7 Man United
BEST STATS
WHAT NEXT?
Man United's attention will now switch back to the Premier League, with the Red Devils facing a trip to Aston Villa on Sunday afternoon.
Porto, meanwhile, will be aiming to make it four straight wins in Portugal's top flight when they resume their domestic campaign against Braga on Sunday night.