The Mexican, who came on to score a dramatic equaliser at Stamford Bridge in 2012, repeated the trick for the Irons with a driven finish into the bottom corner after 73 minutes.
For much of the first half, the east Londoners did look second best, with Chelsea seemingly more risky with every attack.
After a series of mediocre results, a ruthless Spurs side punished Chelsea's inability to capitalise on their first half chances with a 3-1 win at Stamford Bridge, the first time a Tottenham side has won at the ground in 28 years.
Chelsea were booed off at full-time and, asked if they could still make the top four, Conte said: "I don't know".
His introduction to the game was certainly key for West Ham to get their goal, with Arnautovic struggling as a lone striker.
Antonio Conte's side had dominated most of the game, cutting through a soft West Ham midfield seemingly at will, but failed to add to Cesar Azpilicueta's first-half strike before David Moyes's game-changing substitution. This is not the first time and we have to improve a lot.
For 73 minutes, West Ham lacked any sense of adventure, and this point of enormous value in their bid to remain in the Premier League owed much to Hart's refusal to be breached in the second period.
"Coming to Chelsea games as a kid, joining the club at seven and being released at 14, I don't think I need to prove anything, but today I felt I came back and made a statement and I was happy with my performance."
"It's another result which isn't positive because we are talking about a draw".
"It was a good weekend for us because they were singing my name, then I scored and I could celebrate with them".
But next Monday they host fellow strugglers Stoke knowing a defeat would drag them right back into trouble ahead of a tricky run-in including fixtures against Arsenal, Manchester United and Manchester City. I don't understand why after 1-0 we always drop and let the opponent get more ball possession.
Charlie Austin came off the bench to level for the Saints 17 minutes from time. If West Ham can win one or two of those matches, they could be blowing bubbles in the Premier League for another season.