The Gunners lead on goal difference after United were held 0-0 at home by Blackburn.
Arsenal manager Arsene Wenger said: "It's been an interesting day for us. The good thing is that our Champions League situation is in our own hands.
"We made it quite comfortable. Maybe we were cruising too much at 2-0. Then Darren Huckerby scored a great goal and we had to get the third.
"But I thought we always looked like scoring another. France gave Thierry a rest and we benefited today."
Henry took his Premiership total to 25 goals with two trademark curling shots and an opportunist goal. Freddie Ljungberg added Arsenal's fourth as the best attack in the Premiership ripped open the leakiest defence.
Henry passed a scan on a calf injury and came through training on Friday in time to play. How Norwich must have wished he had been sidelined.
The result emphasised the vast gap between the top of the Premiership and the bottom. Norwich were outclassed. They allowed Henry too much space and he embarrassed them.
Norwich boss Nigel Worthington said: "We stuck at it, but Arsenal's quality simply came through. We had two players on Henry for the first goal and he just passed the ball into the net. He was in demolition mode today."
Norwich's plight is desperate. They remain bottom, seven points behind the 17th-placed team with seven games to go. They have lost their last five matches and kept only three clean sheets in the Premiership. They have conceded 63 goals.
They have won only three games, all at Carrow Road and scored just four goals in their last 14 league outings.
They have made one major signing, striker Dean Ashton from Crewe in January. But Delia Smith and the Norwich board have refused to gamble the financial future of the club on staying in the Premiership for another year. They will regroup and chase promotion again next season.
Worthington said: "One of the things we've had to learn to do this season is to lose. That's been an experience in itself, but we've grown mentally stronger as a group."
Arsenal welcomed back Brazil midfielder Gilberto after a seven-month absence with a fractured vertebrae. He replaced skipper Patrick Vieira who was feeling the effects of France's World Cup qualifiers.
Wenger will be delighted when they can resume their partnership. They have hardly played together all season and Arsenal have suffered accordingly.
This was a good game for Gilberto to return. He was able to ease into the pace of the game as Arsenal dominated throughout. Wenger added: "His return is good for us and he looked sharp for a guy who's been out for so long."
The movement of Henry and Jose Antonio Reyes quickly troubled the Norwich defence. Centre-back Craig Fleming blocked a second-minute volley from Reyes. Four minutes later, Henry unleashed a 30-yarder on the turn that was well held by Robert Green.
Henry slipped Reyes through inside the box. Green came to Norwich's rescue again with a dive to palm away Reyes' shot. Reyes lashed a 25-yarder wide, before Henry made the breakthrough in the 19th minute.
Robert Pires threaded through a neat pass. Henry sidestepped two defenders and buried a low curler past Green.
Henry added his second three minutes later. He cut into the box, beat two defenders again and bent another shot beyond Green. The Norwich keeper has been their best player this season. He has had plenty of days like this to show his calibre.
Arsenal's only early scare came when Pascal Cygan misplaced a clearing header but the visitors lacked players forward to capitalise.
Reyes continued to pepper the Norwich goal and Gary Holt blocked another Henry effort.
Then Norwich scored a 30th-minute goal out of nothing. Cygan and Kolo Toure failed to clear Green's long kick. Huckerby cut in from the left and swept a shot past Jens Lehmann. It was only Norwich's seventh away goal of the season.
It was the first goal Arsenal have conceded since Peter Crouch's equaliser at Southampton on February 26, but their central defence still looks vulnerable. Wenger will be relieved to have Sol Campbell back after a long-term ankle injury. Campbell is expected to have a reserve run-out on Monday.
Gilberto drove forward at the start of the second half to shoot over the bar, before Ljungberg's shot was cleared by Damien Francis.
Norwich succumbed again in the 50th minute. Lauren broke to the by-line and chipped for Ljungberg to score from close range. It was that collector's item among Arsenal goals - a header.
Green held another Cesc Fabregas strike as Norwich teetered on the brink of collapse. The keeper turned over Henry's fierce 20-yarder for a corner then held another Reyes drive. He will attract attention from bigger clubs during the summer.
Henry completed his inevitable treble in the 66th minute. Reyes was tackled as he burst into the box. Henry reacted fastest to knock in the loose ball.
Ljungberg shot wide and Green denied Pires as Norwich's penalty area resembled a shooting gallery.
Worthington made three substitutions, but his side's afternoon had been reduced to damage limitation. Lehmann even denied Norwich the consolation of a second goal when he saved Huckerby's volley.
Wenger used the closing minutes to give Robin van Persie and Gael Clichy more game time. Arsenal will rarely enjoy a more comfortable afternoon, Henry had seen to that.
Man of the Match: Thierry Henry - Norwich allowed him space and he punished them with deadly finishing for a clinical hat-trick.