The Red Devils secured a 2-0 win in West London thanks to goals from Ander Herrera and Pogba, with a quarterfinal tie against Wolves to look forward to now on March 16.
Caretaker manager Ole Gunnar Solskjaer has now overseen 11 wins in 13 matches across all competitions and his squad has managed to come through their biggest test to date.
After suffering a damaging 2-0 first leg defeat against PSG at Old Trafford in the Champions League last week, all eyes were on United on Monday night to see if they could respond positively and they rose to the occasion admirably.
Under former boss Jose Mourinho, United ended the 2017-18 campaign without a trophy and Pogba is keen to avoid a repeat of that scenario come May.
"All the competitions are important. We want to win something.
"Last year we didn't win any trophies. I still remember that. This year we want to win something."
United managed to blunt Chelsea's attack while finding space in the final third between the lines on a consistent basis, executing a perfect gameplan set out by Solskjaer.
Pogba, who now has 14 goals to his name across all competitions, went on to offer an insight into their tactical thinking on the night, while also insisting that the team could have been even more effective at certain points during the game.
"We knew how to hurt them, (by) running behind their midfielders," Pogba added. "Two runs in the box, two goals. My job as a midfielder is not to score goals, it's more to make assists. But when I get a chance to score goals it's a great feeling.
"The result was great, but we should do better. We should react even better. We made it harder for ourselves. We can control more, play more and hurt them more with the ball."
Next up for Solskjaer's side is a huge Premier League showdown against Liverpool at home on Sunday, which could see United deal a bitter blow to their arch-rivals in the title race.