The Hammers grabbed the lead in the opening 10 minutes when Michail Antonio converted a cross from Pablo Fornals for just his fourth goal of the season.
West Ham have missed the mobility and presence up front of Antonio, who was sidelined before Christmas with a hamstring injury and has been eased back into action by manager David Moyes given his importance to the team. He came on as a substitute against Southampton and Everton over the festive period, and played in the win over Rochdale in the FA Cup on Monday.
Moyes has few other alternatives for the striker role, especially with Sebastien Haller recently leaving to join Ajax, so keeping Antonio fit is particularly important if West Ham is to finish in the top half of the standings.
Burnley thought they had equalised before half-time when a cross by Chris Wood was turned into his own net by West Ham defender Aaron Cresswell, but the goal was disallowed because Wood was offside in the buildup.
While West Ham are unbeaten in its last four league games, keeping three straight clean sheets in the process, Burnley have lost three of its last four games and dropped to fourth-to-last place, one above the relegation zone.
Scoring is their biggest problem — Sean Dyche's team have just nine goals in 17 games, tied for the fewest in the league with last-placed Sheffield United.
Additional reporting by Reuters.