Solskjaer's men remain in fifth, but moved back level on points with Leicester, who they face on the final day of the season
Ole Gunnar Solskjaer told Manchester United to enjoy the pressure of fighting for a place in the top four after needing a 19-game unbeaten run just to give themselves a chance of a return to the Champions League next season.
United rarely hit the heights of recent weeks in beating Crystal Palace 2-0 on Thursday as goals from Marcus Rashford and Anthony Martial, allied to a helping hand from VAR, kept a top-four finish in their own hands.
Solskjaer's men remain in fifth, but moved back level on points with Leicester, who they face on the final day of the season, and to within one point of third-placed Chelsea.
"We need to smile, we have earned this right. No-one thought in January that we would be in with a shout for top four," said Solskjaer.
"These are the games that matter. That is the thing at Manchester United, you have to step up and show your personality."
Solskjaer bemoaned having 48 hours' less preparation time than Chelsea for Sunday's FA Cup semi-final, but that made little difference to the Norwegian's team selection with only two changes from the side that started Monday's 2-2 draw with Southampton.
Fatigue seemed to play a factor in a slow start from the visitors and Solskjaer warned his side they cannot afford a repeat in a big 10 days ahead.
"Of course it's tight scheduling but we can't afford starts like that," he added. "We didn't get the rhythm in our passing but two fantastic goals won it for us."
Palace have now lost six games on the spin, but Roy Hodgson believed his side did not get what they deserved, particularly with two controversial VAR reviews.
The game turned in the minutes before the break when Wilfried Zaha went down in the penalty area claiming to have been tripped by Victor Lindelof.
Neither referee Graham Scott nor a review awarded a penalty and moments later United led.
Bruno Fernandes fed Rashford inside the area who sold Patrick van Aanholt a dummy and cut onto his right foot before finding the bottom corner for his 22nd goal of the season.
"Is the VAR broken?" tweeted Palace chairman Steve Parish.
Unfortunately for Parish and Palace, VAR was fully in operation 10 minutes into the second half when Jordan Ayew converted Zaha's low cross but was denied an equaliser by a millimetric offside decision on review.
"If there was ever a game I should be sat here congratulating the team on the performance and result tonight is the night," said Hodgson.
"Having to come to terms with something you don't deserve is very difficult."
At the other end Rashford did not get enough power on his shot to beat Vicente Guaita and Fernandes smashed against the post.
Rashford is to become the youngest person to receive an honorary doctorate from the University of Manchester for his work campaigning against child poverty and he rounded off another fine week for the 22-year-old on and off the field with a major role in Martial's second to clinch a vital three points.
The England international burst through three Palace players before feeding Fernandes. The Portuguese's pass through the legs of Luka Milivojevic found Rashford again on the edge of the area and he cushioned a pass into Martial to slide home.
Palace's misfortune was rounded off as Van Aanholt had to be stretchered off after dislocating his shoulder in a collision with Martial, which Hodgson confirmed could see the Dutch international miss the start of next season.