Isaac Mbenza cancelled out Scott McTominay's first-half opener for United, who twice hit the crossbar through Paul Pogba.
United have taken eight points from their last eight games to let a top four spot slip through their fingers a miserable reaction to the full time appointment of Solskjaer.
"Effort and attitude was not the problem, I can say that", claimed Solskjaer. "The players were running and trying". Over a long season we haven't been good enough to challenge for the Champions League places. That is the problem. We could have, should have played better. "It's not come as a surprise but we hoped to show a better performance today".
"Sometimes it happens, concentration, application, confidence drops, it was just odd how we changed our approach to the game instead of playing simple. It is a fair reflection, being sixth".
"We've ended up with the last hope being taken away".
"We're many levels (behind) at the moment, that's the size of the challenge". We weren't able to.
Theresa May tells Labour leader Jeremy Corbyn: 'Let's do a Brexit deal'
The British Prime Minister is trying hard to gain opposition support to get her Brexit deal through parliament this time. The Conservatives lost 1,334 councillors, while Labour failed to make expected gains, instead losing 82 seats.
"We are where we are for a reason", the Norwegian told Sky Sports. "We have to make something positive out of it, no matter what".
Solskjaer was referring to the possibility of his side having to play in a Europa League qualifier should Watford beat City in the FA Cup final. Supporters aghast at a surrender of points to relegated opponents, costing them a chance of qualifying for the Champions League, simply delivered a powerful, voiceless protest that underlined why United need a clear-out in the dressing room and a change of culture in the boardroom.
Lossl made amends by pushing Pogba's header on to the crossbar shortly before half-time, but Huddersfield were allowed to force their way into the game in the second period.
On the hour, Lossl caught a United corner, punted the ball accurately upfield where Shaw failed to clear and Mbenza raced through to push the ball through De Gea's legs.
Town enjoyed good fortune when a late Pogba shot hit the bar once more but might even have enjoyed a ideal final home game when De Gea's legs were all that stood between the United net and an 89th minute victor from Grant.
Solskjaer said his players were not letting him down, but were "a exhausted team", while he refused to be despondent.
Prince Harry Cancels Trip Amid Meghan Markle's Royal Baby Watch
Harry had been due to visit the country for two days from next Wednesday, but will now fly in and out on Thursday. Meghan Markle and Prince Harry are due to welcome their daughter or son any day now.
"I'm an optimist by nature and you make the most of it".
The point for Huddersfield ended an eight-match losing run in the Premier League and was a welcome reward for boss Jan Siewert, who replaced David Wagner in January.
"I know Manchester United is something players would love to experience", he said.
"It shows us where we are".
Huddersfield fans, already well resigned to their fate, were in fine voice in paying tribute to chairman Dean Hoyle, who will step down after agreeing to sell the club. "I am really sure it will go on".
Two dead in shooting at North Carolina university campus
The sad events on the campus of UNC Charlotte have shaken our state. "It's the worst feeling when that text comes", she said. Putney said the handgun used in the shooting was purchased legally. "We're torn up about what's happened", Harden said.
"He is an incredible person", Siewert added.