The Hammers team raised its score to 58 points, ahead of Tottenham VI (56 points) and continued to pressure Chelsea IV (61 points).
Burnley took a penalty kick by Chris Wood in the 19th minute, but the visitors quickly responded with Michelle Antonios goal in the 21st minute.
The latter scored again in the 29th minute, giving West Ham the lead.
Burnley remained at 36 points and is 16th.
Burnley will visit Fulham on May 10, while West Ham hosts Everton on Sunday.
West Ham is seeking to return to the European League, which it has missed since the 2016-2017 season, and it will achieve that if it ends the season fifth.
If he finishes fourth season, he will participate in the Champions League for the first time in his history.