United looked to have snatched a point when Amad Diallo's deflected effort cancelled out Danny Welbeck's first-half opener for Brighton.
But Brazilian forward Pedro was given acres of space to head in at the back post to give Brighton's 31-year-old boss Fabian Hurzeler a dream home Premier League debut.
United have spent heavily in the transfer window to reinforce their defence with the signings of Noussair Mazraoui, Matthijs de Ligt and Leny Yoro.
De Ligt was one of those at fault for the winner as the Dutch international and Scott McTominay were caught under the flight of Simon Adingra's cross to pick out Pedro.
"If you protect the goal like we do then you drop points," said Ten Hag.
"We concede two soft goals where we should act better as a team. We have to be alive in such moments."
Defeat puts the spotlight back on Ten Hag with Liverpool set to visit Old Trafford next weekend.
The former Ajax boss survived an internal review at the end of last season, despite finishing eighth in the Premier League, thanks to winning the FA Cup.
United got a much-needed 1-0 win against Fulham to open the season last weekend thanks to Joshua Zirkzee's late winner.
But the performances in both their first two league matches of the new campaign have done little to suggest there will be a marked improvement from a dismal season in 2023/24.
Emotional win for Hurzeler
Brighton thrashed Everton 3-0 away in Hurzeler's first match in charge and showed they will be a contender to get back into European football next season.
"It was very emotional, always when you get the win in the last minute, it is. I think we deserved to win, we had a good game not a perfect game. There were also situations where United had the chance to win the game," Huerzeler said.
"I'm very proud of the team, they showed the value of never giving up and that's very important to me.
"I think the first half was equal and then in the second half we started to control the game quite well. We had a lot of chances, especially in the second half."
The Seagulls took the lead on 32 minutes when Harry Maguire failed to cut out a dangerous cross and Kaoru Mitoma rolled the ball back across goal for Welbeck to tap in for his 100th career goal.
Within seconds the visitors thought they were level when Marcus Rashford bundled in at the back post only to be denied by the offside flag.
Brighton flew out of the traps after the break but failed to build on their advantage.
James Milner had an effort cleared off the line by Diogo Dalot before Welbeck headed off the bar.
United had barely been in the Brighton half since the restart until Diallo drove forward, cut inside onto his left foot and found the net via a deflection off Jan Paul van Hecke.
Zirkzee had replaced Mason Mount at the break but after a hero's debut, the Dutchman was the villain for what should have been a second United goal.
In a blistering counter-attack, Diallo fed Bruno Fernandes and his low cross was fired goalwards by Garnacho.
However, the ball clipped Zirkzee's knee on its way in and the forward was ruled offside after a VAR review.
That proved to be the turning point as it was Brighton who got the decisive third goal.
Pedro easily pulled away from McTominay and substitute De Ligt to find the space to head back across Andre Onana into the far corner.
"Their goal, for me, was from out of nowhere. I then didn't like the losing the control. After their offside goal I think we started to control the game again. In the end, I think we deserved to win," Huerzeler added.
"I think Joao is a player who can decide the game with one action. For me it was impressive today how he worked against the ball. He had the belief he would score and that's something that makes him special.
"But in the end it's always a team win and that's most important."