United advanced to the May 26 showdown with Villarreal in Gdansk 8-5 on aggregate despite losing 3-2 to Roma in their semi-final second leg at the Stadio Olimpico.
And now his players have to face the prospect of four Premier League fixtures in 10 days after Edinson Cavani's brace staved off a valiant attempt at a comeback from their Italian opposition.
On Sunday they travel to Aston Villa and by next Thursday they will have hosted Liverpool and Leicester, the visit of Fulham coming the following Tuesday.
"It's unheard of. It's made by people who have never played football at this level," raged Solskjaer to BT Sport.
"It's physically impossible for the players. We have not been dealt a good hand. We are going to need everyone for these four games. It's a short turnaround but we have to be ready."
The Norwegian coach was also angry about his side's display on Thursday, admitting that had it not been for a sensational second half in the first leg which gave them 6-2 lead they could very well have been dumped out by a spirited Roma team who created, and wasted a hosted of opportunities in Rome.
"The second half today was disappointing, very, very poor, but we're in the final," he said.
"We played one very, very good half at Old Trafford which has taken us through.
"I'm disappointed we lost, especially the way we played in the second half.
"It could easily have been 8-6 to them. It was a strange game. We kept giving them the ball but luckily we have one of the best keepers in the world. We are in the final and are looking forward to 26 May."
Cavani's goals in the 39th and 68th minutes ensured United's place in the final, the Uruguayan having also struck a double at Old Trafford.
Roma had gone ahead on the night midway through the second half with goals from Edin Dzeko and Bryan Cristante in a three-minute spell, and Alex Telles' own goal gave the hosts a second-leg win with seven minutes to go.
Roma battled from the outset with Gianluca Mancini forcing an early save from United goalkeeper David De Gea, and Henrikh Mkhitaryan nodding the resulting corner into the side netting.
Cavani repeatedly proved dangerous, pouncing on a Roger Ibanez blunder to rattle the bar and then forcing a save from Roma keeper Antonio Mirante.
The Uruguayan finally broke through six minutes before the break, collecting a Bruno Fernandes through ball before firing past a beaten Mirante.
De Gea did well again to save from Mkhitaryan and Dzeko either side of the break.
The 33-year-old Dzeko levelled the match in the 57th minute when he got his head to Pedro's cross to beat De Gea.
Cristante got the second three minutes later, connecting with a Lorenzo Pellegrini to revive Roman hopes.
De Gea then pulled off a miraculous double save on from a Dzeko header and Pedro's follow-up, and Cavani struck again after 68 minutes when he headed Bruno Fernandes' centre past Mirante.
Roma did not give up and Telles' own goal offered some consolation to a team set to be coached next season by former Man United manager Jose Mourinho.
"It was important to come out with our heads held high and have a good game," said Cristante.
"We deserved it, we had a great Europa League this year and it was right to end it like this."