The Hammers played 77 minutes with ten men after Steve Lomas was sent off following a penalty decision. The visitors were also furious at Henry's second goal, claiming that Dennis Bergkamp had fouled Lee Bowyer before crossing for the Frenchman to score with a far post header.
Hammers boss Glenn Roeder said: "We were very resilient but luck was against us.
"Robert Pires didn't need an excuse to fall over. Lomas barely touched him. Then we had the double whammy of the sending-off.
"Bowyer was clearly fouled before the second goal. Bergkamp put his hand in Bowyer's face and pushed him away."Dean infuriated West Ham when he awarded a spot-kick against Lomas for a challenge on Pires - then brandished a straight red card.
The referee insisted that the Hammers midfielder had been the last defender and denied Pires a scoring chance. Henry tucked away the penalty.
Arsenal presented the strugglers with a 39th minute equaliser when Jermain Defoe pounced on Edu's poor back pass and beat David Seaman.
Keeper David James made brilliant saves from Henry, Bergkamp, Sylvain Wiltord and Gio van Bronckhorst before Henry nodded Arsenal's 71st minute breakthrough.
The West Ham players surrounded the referee but he refused to disallow the goal.
Henry completed his treble with a left-foot finish four minutes from time after a swift break by Bergkamp and Pires. It was his 17th goal of the season and made him the Premiership's top scorer.
Henry said: "We passed the ball very well in the second half, after we'd let them back in the game.
"We knew if we kept our cool the chances would come. I don't understand how West Ham are bottom. They really dug in and David James made some great saves.
Arsenal manager Arsene Wenger said: "Subconsciously we lost some of our sharpness when it was 11 against ten in the first half. But the second half was played in the West Ham half and it was just a question of patience."