Antonio set off in pursuit but wouldn't have got near goal had Mee not misjudged the ball. Antonio was gifted possession and rounded goalkeeper Nick Pope to roll the ball into an open net.
Wood eventually salvaged a point in the 85th, rising to nod Johann Berg Gudmundsson's delivery beyond Hart and compounding Carroll's guilt.
After eight matches, West Ham has eight points, five fewer than Burnley which is undefeated in its last six league games.
City open up lead at top
Manchester City have opened up a two-point lead over Manchester United at the top of the Premier League with another display of scintillating brilliance in a 7-2 thrashing of Stoke City at the Etihad Stadium.
After the high-noon showdown between Liverpool and a United side lacking in any great ambition ended in goalless stalemate at Anfield, City cashed in with a third straight league match at the Etihad in which they have struck at least five goals.
Pep Guardiola's men became the first team to net 29 goals in their first eight league games of a top-flight campaign in England since Everton in 1894 as some of their main title challengers faltered.
Their sixth consecutive league win moved them to 22 points with United on 20 and Tottenham staying third on 17 after finally earning their first home win at Wembley, 1-0 over Bournemouth with a Christian Eriksen goal.
Sensation of the day came at Selhurst Park where Crystal Palace, hitherto without a goal or point to their name, downed champions Chelsea 2-1 with Wilfried Zaha, returning from injury, scoring a fine winner just before the break.
Arsenal were surprisingly toppled too as Watford came from behind to snatch a 2-1 win with a 71st-minute penalty from Troy Deeney and a stoppage-time strike from Tom Cleverley, which lifted the Hornets to the giddy heights of fourth on 15 points.
The shock defeats left fifth-placed Chelsea and Arsenal both nine points behind leaders City, whose current brand of kaleidoscopic football makes them favourites to regain the title they last won in 2014.
Gabriel Jesus, with a brace, Raheem Sterling, David Silva, Fernandinho, Leroy Sane and Bernardo Silva all found the target in the demolition of Stoke with the side now looking like a proper Guardiola creation.
"We didn't lose easy balls, we play fast and simple. That's why I am very pleased. It is the best performance since I am here," Guardiola told the BBC.
City's dazzle was in stark contrast to their neighbours' safety-first approach in the underwhelming Anfield game during which the only Manchester United highlight was David de Gea's great save to deny Joel Matip.
Liverpool, despite being much the better side, were blunt in attack, failing to convert any of their 19 attempts on goal.
"Manchester United came here for a point and got it," said Liverpool manager Jurgen Klopp with a sigh.
Not that United's manager Jose Mourinho would entertain the idea of it having been a bit of an anti-climactic affair.
"It depends on what is an entertaining game," he said. "One thing is an entertaining game for fans, another thing is an entertaining game for people who read football in a different way.
"For me, the second half was a bit of chess but my opponent didn't open the door for me to win the game."
West Ham and Burnley drew 1-1 while Swansea City forged clear of the relegation zone with only their second league win of the season, a 2-0 victory over Socceroo Aaron Mooy's Huddersfield Town.
