Needing a barnstorming finish to the Premier League season and a Leicester City wobble to have any chance of finishing top for the first time since 2004, Arsenal seemed on course for a third consecutive win when they led 2-0.
Arsene Wenger's team were then rocked by Andy Carroll's quickfire hat-trick for the Hammers.
Laurent Koscielny's equaliser spared third-placed Arsenal a defeat in a pulsating last London derby at Upton Park before West Ham uproot to the Olympic Stadium but they remained 10 points adrift of Leicester with six games left.
"We have to keep going no matter what, and hope. You never know what can happen," said manager Wenger although his priority now seems to be sealing a 19th straight season in the Champions League by finishing in the top four.
Slaven Bilic praised his sixth-placed West Ham team after the game.
"I'm proud of everything. We were spectacular. The only thing I'm not happy about is the result," said Bilic after a second consecutive draw dented his dreams of a top-four finish.
Manchester City, who are fourth, secured back-to-back league wins for the first time since October, a 2-1 home defeat of West Bromwich Albion putting pressure on fifth-placed Manchester United in the battle to qualify for next season's Champions League.
Sergio Aguero and Samir Nasri were on target after Stephane Sessegnon had given West Brom an early lead as City moved four points ahead of United who visit second-placed Tottenham Hotspur on Sunday.
VILLA MAROONED
The relegation trapdoor is not completely nailed down on bottom club Villa but after an eighth consecutive league defeat, 2-1 at home to Bournemouth, they can start making plans for life in the Championship after 28 years in the top flight.
Villa are marooned on 16 points with five games left and the most they can now attain is 31, the haul that fourth from bottom Norwich City have after they went down 1-0 at Crystal Palace.
The former European champions also have a minus-41 goal difference compared to Norwich's minus-22 so only a surreal set of circumstances can save them and the fans that waved white handkerchiefs are not expecting miracles.
"It's a sombre mood, as you would imagine," said stand-in manager Eric Black. "We're not relegated yet but we're not far off."
Newcastle still retain a modicum of hope but it is fading fast after they lost 3-1 at Southampton to leave Rafa Benitez's side second from bottom, six points behind Norwich and two adrift of Sunderland who host Leicester on Sunday.
Goals from Shane Long, Graziano Pelle and Victor Wanyama put Southampton clear and Andros Townsend's reply was scant consolation for the visiting fans who travelled south as Newcastle crashed to a ninth successive away defeat.
"It's important that everybody continues to believe. The anxiety is killing us sometimes," Benitez said.
Swansea City effectively secured their safety as Gylfi Sigurdsson's goal meant they became the first side to beat Chelsea in the league since Guus Hiddink took charge of the champions in December.
Injury-hit Chelsea produced a lacklustre display as their 15-match unbeaten spell under Dutchman Hiddink ended in disappointing fashion in Wales.
Elsewhere, FA Cup semi-finalists Watford and Everton drew 1-1.
The focus of attention now switches to Leicester on Sunday with the runaway leaders, who are chasing their first top-flight crown, seven points clear of Spurs.
(Editing by Tony Jimenez)
