'We want to get to the final,' Solskjaer told reporters after the win at Stamford Bridge
Manchester United's Ander Herrera and Paul Pogba headed first-half goals to claim a 2-0 win that dumped FA Cup holders Chelsea out of the competition and secured a place in the quarter-finals in an intense tie at Stamford Bridge on Monday.
Spaniard Herrera finished off a fine move with a precise header from Pogba's cross just past the half-hour and United's French midfielder scored himself when he nodded powerfully home from a Marcus Rashford delivery before the break.
It was a measure of revenge for United, resurgent under interim coach Ole Gunnar Solskjaer, who lost to Chelsea in last May's Wembley final under their former boss Jose Mourinho.
"It is a beautiful win," France midfielder Pogba, who had been sidelined under Mourinho, told the BBC.
"Today was a great performance from the team, all of them... The team helped me get to my best. I give everything for them. They help me score goals and get assists. We are enjoying playing together."
After the game United were drawn to face Wolverhampton Wanderers at Molineux in the quarter-finals.
"An away tie again after Arsenal and Chelsea... We'll have to do it the hard way, Wolves are a tough team to play against," Solskjaer said.
"The performance tonight was fantastic, our tactics worked. We asked our midfielders to press higher and get into the box and it worked. The defending in the second half was fantastic."
The fifth round defeat piled more pressure on Chelsea coach Maurizio Sarri, whose side were hammered 6-0 last week in the Premier League by United's neighbours ManchesterCity. Chants of "sacked in the morning" broke out in the second half.
United's Sergio Romero had pulled off a fine double save in the 11th minute from a spinning David Luiz free kick headed for the top corner followed by a stinging shot from Pedro.
Three minutes later Arrizabalaga had to fly to his left to keep out a Herrera strike. The lively Rashford also kept the Chelsea keeper under constant pressure.
Chelsea's fragile defence was split with crosses from either flank. Pogba supplied the first in the 31st minute after a fast move down the left and Herrera ghosted into the box to nod the ball down past stranded goalkeeper Kepa Arrizabalaga.
Until that moment a lively game had been evenly poised, with a host of chances at either end.
Chelsea's hopes were crushed just before halftime in a speedy United counter down the right, with Marcus Rashford crossing for Pogba to head powerfully home between defenders Cesar Azpilicueta and Antonio Rudiger.
United were able to drop back and defend after the break, making occasional forays up the pitch but always confident of keeping an increasingly toothless Chelsea attack at bay.
The home side’s best chance of the half came in the 63rd minute when Eden Hazard, reportedly a target for Real Madrid, jinked his way through, only for the Belgian's shot to be expertly diverted by United defender Victor Lindelof.
The home fans jeered Sarri’s like-for-like substitutions. The Italian has been criticised for stubbornly sticking to his tactics and failing to change games.
United, on the other hand, looked like a team on a mission, playing for their coach and each other and brimming with the confidence that comes from turning an ailing season around.
Their only reverse since Norwegian Solskjaer took over was last week's Champions League home defeat by Paris St Germain.
The contrast was reflected in the boos from some Chelsea supporters at the final whistle, while a jubilant Solskjaer celebrated with his players in front of the travelling fans.
"We want to get to the final," Solskjaer told reporters after the win at Stamford Bridge.
"It'll be a tough away game against Wolves, but it seems like both we and our fans love the away games. It didn't feel like an away game with that support behind the goal.
"We lost the final to Chelsea last year. My last game (as a player) was a loss to Chelsea in the FA Cup final. Last time we beat them in the cup was 20 years ago, apparently."
Fourth-placed United will shift their focus back to the league and the race for the top four this weekend when they host title contenders Liverpool, with both Arsenal and Chelsea a point below them in the standings.
"We play Liverpool next week, but Chelsea, Arsenal and us will fight for that fourth position... three good teams fighting for that one position. But this was about getting through to the next round," Solskjaer added.