The Gunners are two points behind second-placed United who were held 0-0 at Crystal Palace. Meanwhile, Portsmouth are sliding down the table fast after winning won only once in their last 11 Premiership games.
Pompey coach Joe Jordan warned before the game that they needed points to avoid being sucked into the relegation dog-fight. Now they are only six points ahead of local rivals Southampton, who occupy 18th place.
Henry's treble - a shot, a chip and a curling free-kick - gave the Gunners' confidence a crucial boost, for a week that manager Arsene Wenger believes is the most important of their season so far.
Arsenal face Bayern Munich in the Champions League last 16 on Wednesday - trying to overturn a 3-1 first leg deficit - then go to Bolton next Saturday in the FA Cup quarter-finals.
Wenger said: "Thierry creates goals as well as scores them which makes him so important to us. It's natural that a striker can't always score but his finishing was excellent today. The keeper touched Thierry's shot for his third goal, but I believe goals like that should be given to the guy who takes the free-kick."
He added: "This was a very solid team performance, very focused, full of team spirit and resilience. We had to be steady and patient. It became a comfortable win but we had to work hard to make it comfortable.
"We fielded a young side which showed great maturity. The experienced players led the younger ones well. It was an interesting mixture."
Yet the Gunners still showed defensive weaknesses that might have been punished by stronger opposition than Portsmouth.
Wenger's squad have been feeling the physical effects of a gruelling season and he admits that fatigue has become a serious worry. Long-term injuries to Sol Campbell, Gilberto and Edu, plus suspensions for Dennis Bergkamp and Jose Antonio Reyes have prevented him rotating his players.
Robert Pires missed this game with an ankle injury and is doubtful for the Champions League game against Bayern. Defender Pascal Cygan came off injured in the 23rd minute, to provide another headache for the Arsenal boss.
Wenger rested Lauren and Freddie Ljungberg from his starting line-up. Henry, who had recovered from an Achilles problem, was partnered by Dutch teenager Quincy Owusu-Abeyie, while teenager Ces Fabregas continued in midfield.
Patrik Berger returned for Portsmouth after flu but ex-Gunner Richard Hughes was ruled out with a pulled hamstring.
They started with Yakubu Ayegbeni as a lone striker in front of a five-man midfield, with Lomana LuaLua operating as a breaking midfielder in support.
After five minutes the DR Congo international sprang the Arsenal offside trap and fired a 20-yard effort that Jens Lehmann held.
Quincy flashed a 20-yard shot across goal as the Gunners responded before Henry, checking his range, dribbled past two tackles and shot wide from the edge of the box.
Gary O'Neil gave the Gunners a scare when he nipped past Cygan before Lehmann cut out the cross. Yakubu came even closer in the 14th minute, out-jumping Cygan to meet Matthew Taylor's centre with a header that thudded off the bar.
Bayern's spies will have noted Arsenal's continuing vulnerability in the centre of defence. So will Bolton, who are one of the Premiership's set-piece experts.
A free-kick, immediately followed by a corner, caused more chaos in the Arsenal box before Arjan de Zeeuw was penalised for fouling Lehmann. De Zeeuw later rose above the home defence to head over Berger's corner.
Henry finally broke down Portsmouth's resistance in the 39th minute, meeting substitute Lauren's cross with a decisive low shot.
Patrick Vieira wasted a fine chance to add a second as half-time approached, blazing over from close range after Konstantinos Chalkias fumbled Henry's corner.
In between, O'Neil's cross hit Mathieu Flamini's hands but referee Chris Foy waved away Portsmouth's penalty appeals, while Linvoy Primus and Dejan Stefanovic were booked for fouls.
Philippe Senderos climbed higher than the Pompey defence for Henry's 49th-minute corner but could not direct his header.
Then Henry struck again in the 53rd minute. Vieira won the ball and timed his pass superbly. Henry ran on, waited for Chalkias to commit himself, then coolly chipped him from 15 yards. Bayern will be eager to stop the Premiership's top scorer running with such freedom.
Portsmouth still occasionally threatened and Lehmann saved smartly from Steve Stone's shot at a Taylor corner, after Arsenal had conceded the kick after a mix-up between the keeper and Kolo Toure.
The lively Quincy forced Chalkias into action once more with a 20-yard drive before he was replaced by Robin van Persie, while some of his colleagues seemed to drop down a gear with an eye on the big games ahead.
Yakubu almost made them pay in the 73rd minute, shooting wide from close range.
But Henry finished the contest, completing his hat-trick five minutes from time. He was brought down by de Zeeuw on the edge of the box and then curled the free-kick into the net, although Chalkias got his fingers to the shot.
Chalkias must be sick of the sight of the Frenchman as Henry scored against him earlier this season when he was playing for Panathinaikos in the Champions League.
Portsmouth have still never beaten Arsenal in the Premiership and Jordan admitted they deserved nothing this time round.
"I was disappointed today. Last week (against Manchester United) I thought we deserved something from the game," said Jordan.
"When you play well and don't get any points you can take some degree of satisfaction from the performance. But today, over the 90 minutes, I can't say that we deserved anything at all.
"We have to look at our own destiny. We can't affect the teams below us unless we play them. But now we face Newcastle and we have to concentrate on that."
MAN OF THE MATCH: Thierry Henry
A master class in finishing with three fine goals.