Palace completed a first-ever league double over United in style as Michael Olise scored twice, while Jean-Philippe Mateta and Tyrick Mitchell were also on target for the in-form Eagles.
Defeat left Ten Hag's men eighth and at risk of both their worst-ever Premier League finish and missing out on European competition next season.
The Dutchman's future is in doubt with structural change ongoing at Old Trafford since British billionaire Jim Ratcliffe bought a minority stake and took sporting control of the club earlier this year.
"It’s clear and it's obvious this is under-performing," said Ten Hag.
"We didn't act how we want to do it and this is by far not good enough."
Ten Hag's options were again limited by a mounting injury list.
Captain Bruno Fernandes missed a club game through injury for the first time in his career, while England international duo Marcus Rashford and Harry Maguire were also absent.
The makeshift centre-back pairing of Casemiro and Jonny Evans were torn apart by Palace's pace and invention.
"Absolutely," added Ten Hag on whether he believes he is the right man to lead United forward.
"If we have the right players available we have a good squad but we miss almost the whole back-line and then we have problems.
"I will keep fighting and I prepared the team in the best way I could do. It was not good enough, by far not good enough, so I have to take the responsibility for that but I will find the energy and I will prepare them for Sunday."
Worse could be to come for Ten Hag when title-chasing Arsenal visit Old Trafford at the weekend.
United have never failed to finish in the top seven in the Premier League era and must do so or beat Manchester City in the FA Cup final to qualify for Europe.
But the embarrassing scoreline at Selhurst Park is not unprecedented during Ten Hag's reign.
United lost 4-0 at Brentford, 7-0 to Liverpool and 6-3 away to Manchester City during his first season in charge.
This season they have conceded 81 goals in all competitions, the club's worst tally since 1976-77.
After disappointing early exits from both the Champions League and League Cup, an FA Cup run has offered Ten Hag some relief.
But they booked their place in a second consecutive final only by surviving a penalty shootout against second-tier Coventry after blowing a 3-0 lead to draw 3-3.
"That was tough to watch," said former United midfielder Paul Scholes of the Red Devils' display at Palace.
"That could have been 7-0."
Ratcliffe's INEOS group have already begun making appointments they hope will turn around a decade of decline at the 20-time English champions.
Jason Wilcox has been named the club's new technical director, while Newcastle's Dan Ashworth has been targeted as sporting director and former Manchester City executive Omar Berrada will take over as CEO.
