The former England international struck his 20th goal of the season from close range following a dreadful back pass from John Oster which enabled Marlon Harewood to round home keeper Danny Coyne and square the ball for Sheringham to score.
In a game of few chances Sheringham was the difference between the two sides and single-handedly took it upon himself to try and win the game.
After firing wide from 20 yards Sheringham sent a ferocious header from a Mark Noble corner wide of the near post before seizing upon Oster's awful back pass.
Hammers boss Alan Pardew was full of praise for his two strikers, saying: "I don't think I have seen the front two look as good as they have in the last two games.
"They seem to have come together at this stage in the season and I hope that this continues. They linked up really well tonight and there is a nice balance between the two.
"The one good thing about them is that they both respect each other. The only disappointing thing for me was how long Teddy took to put the ball in as I was a bit nervous!"
Burnley boss Steve Cotterill was left flabbergasted: "I didn't actually see the goal. I was talking to Michael Duff and Danny Coyne had the ball in his hands.
"The next thing the ball was in the back of our net. It was just a stupid goal to give away. "
Cotterill thought that Sheringham was the difference between the two sides: "You don't want him to have the ball six yards out because there is only one outcome.
"You know he is going to score and you would rather it had been someone else as they might have panicked, but he was very cool and slotted it away."