Spurs bossed the first half with long periods of keep ball. Robbie Keane thumped home the only goal from the penalty spot after being pulled down by Alan Stubbs.
Anthony Gardner, Michael Carrick and Keane all had chances to score only to find Everton keeper Richard Wright in good form who saved his best for Teemu Tainio's header.
Everton started the second half brightly and Ledley King had to be alert to block from Tim Cahill. But Spurs continued to waste chances as Wright had to make another save to deny Jermaine Jenas before somehow keeping out Jermain Defoe's close-range effort.
Martin Jol's men had to withstand a late barrage from Everton as Duncan Ferguson was sent on to give his own brand of attacking power to Everton after 65 minutes, replacing Simon Davies, but Spurs held on.