The visitors were fortunate just before halftime when VAR ruled out Son Heung-min's goal for offside, but their luck deserted them after the break at an empty Tottenham Stadium.
Defeat left West Ham above the relegation zone on goal difference from Bournemouth who have a game in hand.
Tottenham's first win in eight matches in all competitions kept them in the hunt for a Champions League qualification berth.
Jose Mourinho's side are seventh with 45 points, one behind fifth-placed Manchester United.
Brighton are six point above the drop zone after scoring a 0-0 draw at third placed Leicester.