Deep into second-half stoppage time a cross from a corner kick grazed Shaw's outstretched arm and, after consulting the pitchside monitor, referee Andre Mariner gave Brighton a penalty that Alexis Mac Callister converted to give his side all three points.
"Last minute, last action really. I got a little nudge, but of course the hand shouldn't be there," Shaw told Sky Sports after the game. "I own up to it, take it on the chin, it cost us the game, but it is tough. I can't even explain why my hand is up there.
"I got a nudge, maybe that helps with my hand going up, but it shouldn't be there."
The result may have felt like a bit of revenge for Brighton after their loss to United in a penalty shootout in the FA Cup semifinals 12 days earlier. It moved Roberto de Zerbi's side into sixth place in the table, eight points behind fourth-place United, but with a game in hand in the race for Champions League qualification.
"It changes nothing. It is still in our hands," Shaw said of the race for the top four. "We have to pick ourselves from the position we were win. We were comfortable but now you could say there is a real chase.
"Teams around us are picking up points. We still have a game in hand and it is in our hands."
The dramatic end of the match brought back memories of a meeting between Brighton and United at the Amex Stadium in 2020, which was decided by a converted penalty from United's Bruno Fernandes in the 10th minute of stoppage time.
This time, Brighton were on the right side of a late penalty, scoring their latest-ever goal in the Premier League and the latest ever against United in the competition.
De Zerbi felt a sense of justice after his team's impressive display at Wembley Stadium last month.
"Both games, I think we played better than United," De Zerbi said.
Just like in their FA Cup meeting, both teams created plenty of chances at the Amex but found the goalkeepers in top form.
At the other end, Brighton's Jason Steele produced almost identical saves to deny Marcus Rashford and then Anthony Martial in the first half before a second half that the hosts edged and forced De Gea into two more game-saving stops before Mac Allister's clinching penalty.
Manchester United play West Ham United at the London Stadium on Sunday, while Brighton face an Everton team in serious danger of being relegated from England's top flight for the first time since they went down following the 1950-51 season.
Information from the Associated Press was used in this story.