Arsenal, Everton and Newcastle United will also be in Monday's draw after seeing off second division opponents, while Southampton needed a stoppage-time penalty to defeat 10-man Portsmouth 2-1 in a south-coast derby.
Arsenal beat Wolverhampton Wanderers 2-0, James Beattie scored his first goal for Everton in a 3-0 win over Sunderland and Newcastle captain Alan Shearer got their first in a 3-1 win over Coventry City.
However, none of the goals could match the double strike by Rooney at Old Trafford, where United made amends for needing a replay to knock out minor league Exeter City in the third round.
John O'Shea had given United a 10th minute lead before Rooney latched onto a Gary Neville pass and lobbed Boro keeper Mark Schwarzer from a full 30 metres out in the 67th minute.
Rooney's second was a candidate for goal of the season after keeper Roy Carroll punted the ball upfield, Frenchman Louis Saha headed on and the England striker volleyed home from 20 metres.
"They are two of the best goals I've scored in my career," said the 19-year-old former Everton forward.
Manager Alex Ferguson was impressed by Rooney's contribution, saying: "Fantastic, particularly the second. The first was precision...marvellous.
"We played with a very good tempo today and I was very pleased with the performance all round."
It needed a penalty to break the deadlock at Highbury, where Arsenal striker Thierry Henry was again instrumental.
The Frenchman, who also had a goal disallowed, won a 53rd minute spot-kick that was converted by captain Patrick Vieira and then set up Fredrik Ljungberg to score their second.
Arsenal manager Arsene Wenger said: "They raised their game in the first half and it took us some patience not to be frustrated... I knew the response would come in the second half and we did that very well."
Everton deserved their win at Goodison Park, helped by Beattie's first strike since joining for six million pounds ($11.29 million)
Shearer's strike for Newcastle was followed by further goals from Shola Ameobi and Celestine Babayaro.
The place to be for drama was St Mary's, where the eagerly-awaited tie between Harry Redknapp's Southampton and the Portsmouth side he quit in November was decided by a Peter Crouch penalty in the fourth minute of stoppage time.
Midfielder Matt Oakley gave Southampton a 54th minute lead, but Portsmouth's Nigerian striker Yakubu Aiyegbeni levelled from the penalty spot just two minutes later.
Portsmouth looked set for a replay despite a 72nd-minute red card for French striker Diomansy Kamara until the hosts were given a hotly-contested penalty for handball by Matthew Taylor.
Fulham were held to a surprise 1-1 draw at Derby County, while fourth division Yeovil, famous for their 20 wins over League clubs as a minor-league team, put up a real fight before going down 3-2 at Charlton Athletic.
Yeovil manager Gary Johnson said: "The whole crowd gave us a standing ovation when we walked off, so I think we were part of a very entertaining game -- which is what we set out to do."
Charlton boss Alan Curbishley admitted: "It was game on for the last 20 minutes. They threw caution to the wind and it favoured them, they got back in it. Great credit to Yeovil."
As for his own side, Curbishley added: "I was really disappointed at halftime and I let them know."
Tottenham Hotspur drew 1-1 at Premier League bottom club West Bromwich Albion, while Blackburn Rovers romped to a 3-0 victory over third division Colchester United.
Second division Burnley, who caused one of the upsets of the last round by knocking out Liverpool, cruised to a 2-0 win over third division Bournemouth.
Premier League leaders Chelsea play Birmingham City on Sunday, while the replays will be played on February 12.