West Ham made it 2-1 in the 68th minute when midfielder Grady Diangana played through Anderson, who fired a low drive past Joe Hart for his second league goal of the campaign.
All Whites striker Chris Wood has broken a long goal scoring drought but his header wasn't enough to earn Burnley a win at Winston Reid's West Ham.
It was nearly 2-0 to the hosts when Anderson's goal-bound effort was miraculously kept out by Ben Mee, while the lively Grady Diangana should have been awarded a penalty after a trip from Steven Defour.
West Ham continued to dominate after the restart and Anderson regained their lead with a fine finish on 68 minutes following good link-up play by Arnautovic and Diangana.
West Ham have been Wood's lucky team since he joined Burnley on a $26 million transfer from Leeds United previous year.
Five minutes later, Wood rose above the Hammers defence to power home a header from fellow substitute Jeff Brady's corner.
Burnley have failed to keep a clean sheet in nine of their last 10 Premier League games, shipping at least twice in six of those contests.
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Chris Wood levelled the tie-up once again ten minutes later before the Brazil worldwide added his second of the game five minutes from time.
"The manager, the players, the staff here... it would be hard for me not to improve".
"We were twice ahead, they drew twice, then we scored the third and then wanted to score a fourth, not just to defend our advantage".
Declan Rice seems to be growing in influence with each passing week and was impressive at the base of West Ham's midfield once again.
"You don't need to continue to play towards your own box if you are level or losing, you must continue to play in the same way, and I think today was a test for the players".
- West Ham's Marko Arnautovic is now the highest goalscorer in all competitions at the London Stadium (nine goals).
The academy product finished the game with six completed dribbles, more than any other player on the pitch.