Spurs' UEFA Champions League hopes were left hanging by a thread after they slipped to a limp 3-1 defeat at Bramall Lane last time out.
Jose Mourinho's men were not much better in the more familiar confines of Tottenham Hotspur Stadium on Tuesday (AEST), but Keane's moment of misfortune proved enough to see off a dismal Everton side.
Captain Hugo Lloris was involved in a half-time flare-up with Son Heung-min and an unconvincing triumph that moves Tottenham within seven points of fifth-placed Manchester United will do little to improve the mood in the camp.
Tottenham struggled to break down its visitors in the opening stages but a 21st-minute effort from Lucas Moura almost brought about the opener as his 25-yard effort skimmed the left-hand post.
Just three minutes later Tottenham was ahead as Giovani Lo Celso's effort took a very fortuitous ricochet off Keane that left Everton goalkeeper Jordan Pickford helpless.
Despite their half-time advantage, there were clear signs of discord in the Tottenham dressing room as an argument between Lloris and Son threatened to boil over as they left the pitch.
The pair had to be separated by team-mates and only the reflexes of Pickford prevented Son from putting further distance between Spurs and Everton after the restart, the England goalkeeper twice denying the Korea Republic star.
Spurs had no problem protecting their lead but the prospect of them catching in-form United with five games to play looks an extremely unlikely one.