Wayne Rooney and substitute Juan Mata scored two goals apiece as the soon-to-be-dethroned English champions left the Canaries just a point above the relegation zone with two games remaining. Giggs, who decided against naming himself in the starting side or on the bench, saw Rooney give United the lead with a 41st-minute penalty after Danny Welbeck went down in the box.
England striker Rooney made it 2-0 seven minutes after the break with a curling shot from outside the box as Norwich's defence backed off. Mata put the result beyond doubt in the 63rd minute when he turned in Phil Jones's cross before adding a second goal ten minutes later as Norwich retreated again.
United great Giggs, previously a player-coach, was put in caretaker charge for the final four games of the season when former manager David Moyes was sacked on Tuesday. Moyes was dismissed just ten months after replacing the retired Alex Ferguson following a 2-0 defeat by Everton -- his former club -- made it mathematically impossible for United to qualify for next season's UEFA Champions League.
Everton remain a point behind Arsenal, who currently occupy the fourth and final Champions League spot, ahead of the Gunners' match at home to Newcastle United on Monday. But Everton manager Roberto Martinez insisted the Merseysiders, whose final two league games of the season are against title contenders Manchester City and FA Cup finalists Hull City, had not given up on a Champions League place.
"Not at all, from our point of view it doesn't change at all," Martinez told BT Sport. "Today was a real setback, but we'll be ready for next Saturday."
Everton fell behind in under a minute at St. Mary's when Antolin Alacaraz headed a cross from Southampton's Rickie Lambert past goalkeeper Tim Howard. And the visitors went 2-0 down in the 31st minute when Seamus Coleman diverted Nathaniel Clyne's cross into his own net.
Fulham missed the chance to climb out of the relegation zone after Hull City came from 2-0 down to secure a 2-2 draw at Craven Cottage. Ashkan Dejagah curled in a 55th-minute shot and the west London side then doubled their lead through Fernando Amorebieta's header three minutes later.
Nikica Jelavic pulled a goal back for Hull in the 75th minute and they equalised three minutes from time when Shane Long headed in Sone Aluko's cross.
"It was very disappointing," Fulham manager Felix Magath said. "If you are leading 2-0 at home, you have to see it through."
West Bromwich Albion moved five points clear of the relegation zone with a 1-0 win at home to West Ham United. Saido Berahino scored the only goal at The Hawthorns when he turned in Morgan Amalfitano's centre in the 11th minute.
"I think that we need to win one of our final three games to be safe," West Brom manager Pepe Mel said.
Wilfried Bony opened the scoring for Swansea before Gabriel Agbonlahor briefly drew visitors Villa level in south Wales. However, midfielder Jonjo Shelvey restored Swansea's lead in the 26th minute with a stunning volley from inside the centre circle that dipped over the head of back-pedalling Villa goalkeeper Brad Guzan.
Pablo Hernandez made it 3-1 before Ivorian striker Bony converted a late penalty for his 24th goal of the season.
Tottenham Hotspur moved to within three points of fifth-placed Everton with a 1-0 win at Stoke City, courtesy of Danny Rose's 33rd-minute header. Stoke were reduced to ten men in the second half when Ryan Shawcross was sent off after receiving a second yellow card for a studs-up challenge on Rose.
The race for the title will intensify on Sunday when leaders Liverpool face second-placed Chelsea, five points behind, at Anfield.
Third-placed Manchester City, six points off the summit but with a game in hand on the top two, are away to Crystal Palace.
At the other end of the table there is a relegation clash when bottom of the table Sunderland host fellow strugglers Cardiff City.