Desperate to win a trophy to rescue a frustrating campaign, United advanced to the semi-finals with a 2-0 win over Brighton at Old Trafford on Saturday.
Tottenham won 3-0 at Swansea to book their second successive appearance in the last four.
Chelsea, beaten by Arsenal in the final last May, are eyeing a return to the title match after winning 2-1 at Leicester on Sunday.
Alvaro Morata ended a personal goal drought stretching back to December 26 as the Spain forward put Chelsea ahead with a clinical finish in the 42nd minute.
Leicester striker Jamie Vardy poked home from close range in the 75th minute to send the quarter-final into extra time.
But Spain winger Pedro won it for Chelsea in the 105th minute when he punished Leicester goalkeeper Kasper Schmeichel's misjudged attempt to punch clear.
Pedro got to the ball before Schmeichel and met N'Golo Kante's cross with a header that looped into the empty net.
The victory kept alive Chelsea's last remaining hope of winning silverware at the end of a difficult season.
Chelsea had lost three of their past four matches to fall behind in the race for a top-four finish in the Premier League as well as bowing out of the Champions League against Barcelona.
"I think Leicester played very well, but it was important for us to win," Chelsea midfielder Kante said.
"It is the only title we can have this season and we have to fight to the end."
- Ideal start -
Meanwhile, Southampton are back in the semi-finals for the first time in 15 years.
New Saints manager Mark Hughes enjoyed the ideal start to his reign as the Premier League strugglers won 2-0 at Wigan on Sunday.
Hughes was hired this week to replace the sacked Mauricio Pellegrino and the former Southampton striker had an immediate impact on his new team.
Hughes' main goal is to help Southampton preserve their top-flight status and they dropped into the relegation zone on Saturday.
Nevertheless, a trip to Wembley in the semi-finals could provide a welcome tonic as Hughes tries to make his presence felt.
It was essential Southampton avoided following in the footsteps of Manchester City, West Ham and Bournemouth -- top-flight sides who had all perished against third tier giant-killers Wigan in the FA Cup this season.
They achieved that aim thanks to second-half goals from Pierre-Emile Hojbjerg and Cedric Soares.
"It was an important day for the players today. I have been really impressed with the quality of the talent I have," said Hughes, who was sacked by Stoke earlier this season.
"They have lost a bit of confidence recently but they are good players.
"We stood up to Wigan. It was a statement of intent and a start.
"We have work to do in the Premier League, but we will enjoy this moment."
The semi-finals will take place over the weekend of April 21-22.