Newcastle also crashed to a shock defeat as Dominik Solanke's double took Bournemouth out of the relegation zone.
Rocked by Wednesday's devastating Champions League group stage loss in Copenhagen -- the latest embarrassment in a season of low moments -- United gave Ten Hag some much-needed breathing space thanks to Victor Lindelof's second-half winner.
United had suffered five defeats in their previous 10 home games this season, their worst start to a campaign at Old Trafford since 1930-31 when they also lost five of 10 and were relegated from the top flight.
They were also reeling from three losses in their last four games in all competitions as the heat was turned up on Ten Hag.
Against that depressing backdrop, and with the latest 'Glazers out' protests from fans outside Old Trafford before kick-off, struggling Luton's first visit to United since 1991 was well-timed for Ten Hag and his troubled team.
Sixth-placed United made their pressure count in the 59th minute as Bruno Fernandes' corner was cleared to Marcus Rashford, whose cross fell kindly for Lindelof to lash home.
"It is a good day. We needed the win and we got it, but we could make life more easy by scoring early on and getting a second," Ten Hag said.
"I have to compliment the team that they kept going, kept organised and didn't give anything away."
Wolves' stunning late show denied Tottenham the chance to return to the top of the table.
Brennan Johnson's first goal for Tottenham since his move from Nottingham Forest put Ange Postecoglou's side ahead in the third minute.
But Wolves dominated after that and earned a memorable victory thanks to the incredible finale.
Pablo Sarabia finally bagged the equaliser in the first minute of stoppage time when he smashed past Guglielmo Vicario from Matheus Cunha's pass.
Missing injured stars James Maddison and Micky van de Ven, Tottenham were well below their best and Mario Lemina made them pay in the 97th minute, meeting Sarabia's cross with a cool finish from 12 yards.
'We ran out of legs'
After starting the season on a 10-game unbeaten run, it was Tottenham's second successive defeat after Monday's 4-1 loss against Chelsea when they had two players sent off.
"Wolves are always going to come on strong and we ran out of legs at the end," Postecoglou said.
"It is understandable, a lot of those guys haven't played. They scored a couple of good goals. It's a hard one to take."
Arsenal took advantage of Tottenham's stumble as they beat Burnley 3-1 to move one point above their north London rivals into second place.
Leandro Trossard opened the scoring with Arsenal's 1,000th goal at the Emirates Stadium since their move from Highbury in 2006.
Burnley levelled in the 54th minute when Josh Brownhill fired home via a deflection off Arsenal defender Gabriel.
But William Saliba's close-range header three minutes later put Arsenal back in control and Oleksandr Zinchenko volleyed the Gunners' third in the 74th minute.
Arsenal were reduced to 10 men in the 83rd minute when Fabio Vieira was shown a straight red card after catching Brownhill on the knee.
An injury-ravaged Newcastle paid for their exertions losing away to Borussia Dortmund in the Champions League in midweek.
Solanke fired home from close range twice in 13 second half minutes to secure Bournemouth's second league win under Andoni Iraola in a 2-0 victory.
Newcastle have now failed to win half of their 12 Premier League games this season and remain seven points off the top down in seventh.
Everton climbed away from the relegation zone with a 3-2 win at Crystal Palace.
Vitaliy Mykolenko, Abdoulaye Doucoure and Idrissa Gueye scored for the Toffees, who move eight points clear of the bottom three.
