Clarets' supersub Chris Wood's late header five minutes after coming on looked like it would lead to a share of the points.
But Anderson struck with just six minutes to go after looking to have scored the goal that would assure his side victory just over 15 minutes earlier, before Javier Hernandez netted in injury time.
The 25-year-old Brazilian was told by his boss to buck up his ideas and he did just that with his 68th-minute strike through the legs of Joe Hart.
Marko Arnautovic gave the Irons the lead ten minutes in when he netted his fifth of the season after pouncing on James Tarkowski’s defensive horror show.
But Burnely – totally outclassed and outplayed in the first half – equalised in the 45th minute through Johann Berg Gudmundsson.
Manuel Pellegrini’s side know they need to pick up points in a fixture list that sees them face Huddersfield, Newcastle, Cardiff, Crystal Palace and Fulham before Christmas.
Manchester City travel to London after the Terriers clash but the rest offer a great chance to boost confidence.
And how they will need Arnautovic to be on fire for them. If today is anything to go by then he should be as he looks to be nearing full fitness.
His goal was sweet revenge after Tarkowski almost wiped him out with a challenge two minutes before and Arnautovic celebrated in the defender’s face.
He nearly got his second ten minutes later after a silky one-two with Felipe Anderson but Clarets keeper Joe Hart did well to keep it out.
Burnley nearly had a second defensive catastrophe when Charlie Taylor dithered on the ball but Grady Diangana failed to rob him and bear down on goal.
Ben Mee pulled off an astonishing diving header to deny Anderson just after the half-hour mark and West Ham should have had a penalty when Diangana was clipped by Steven Defour but referee Roger East played on.
Diangana, a 20-year-old academy product, looked a real danger on the right and through the middle and it was his assist that set Anderson up.
Anderson then had a chance to win it with ten minutes to go but his shot from where he had just netted went wide before he later made amends in style.
Hernandez made it four in stoppage time after putting away Michail Antonio's chested pass to give West Ham the sort of scoreline they deserved.
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