Following a 2-0 defeat to Real Madrid on Thursday, the Red Devils responded by taking the lead through Diogo Dalot midway through the first half.
But a quick-fire brace from Donyell Malen shortly before the break turned things around for the German side.
Antony then levelled things up early on in the second period, before Youssoufa Moukoko bagged the winner with 19 minutes remaining.
And Ten Hag was damning in his assessment of his side’s display – focusing in on a dismal final 30 minutes.
He said: “Yeah, [the game was] two parts, the first hour was good, playing as a team, lot of dynamic, good pressing.
“So it was really enjoyable to see and that’s what I wanted to say at half-time, but all of a sudden, they give two goals away.
“They were giveaways and that was annoying. But the last half an hour, it was bad, a poor performance.
“They didn’t follow the rules at all, not in pressing, not in building up or when attacking. They were 11 individuals on the pitch.
“That wasn’t good at all. I said to the team, I think it was a good tour, but you want to finish with a good performance and there was only 60 minutes that I was pleased with, but not with the goals we conceded because the first one was poor defending and the second one was poor in build-up.
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“That’s not us, we have to keep control, keep responsibility on the ball and it was the same for the 3-2.
“That can’t happen and it’s unforgivable, even if it’s a friendly.”
New keeper Andre Onana – who joined from Inter Milan for £47million earlier this month – was seen berating defender Harry Maguire after his poor pass nearly led to a Dortmund goal.
And the loss is not the best preparation for United – who have two friendlies left ahead of their 2023/24 campaign.
They return to Old Trafford on Saturday to host Lens, before travelling to Dublin to face Athletic Bilbao the following day.
Man Utd kick off their Premier League season at home to Wolves on August 14.