Harry Kane, Ryan Mason and Toby Alderweireld were kept at bay by Howard in the first half, although Kane should have finished his one on one with the American goalkeeper.
Spurs, who named Erik Lamela on the bench, only had themselves to blame for not grabbing all three points in a dominant second half that saw chance after chance squandered.
Both sides had a player carried from the pitch on a stretcher, Everton’s Tom Cleverley in clear pain after being felled by Eric Dier and Mousa Dembele following a bizarre self-inflicted injury.
Tottenham’s dominance was apparent early on as Nacer Chadli dazzled down the left, teeing up Nabil Bentaleb only to see a tame shot scuffed wide of the right post.
There was panic in the Spurs defence when Kyle Walker gave the ball away to Cleverley and it took a fine save by Hugo Lloris to keep out the ensuing shot.
Everton should have slipped behind in the 24th minute but Howard stuck out his left leg to stop Kane, who only had the visiting keeper to beat after collecting a superb pass by Mason.
It was then Mason who was denied by Howard, with Chadli the provider once more and the American then steered an Alderweireld header over the crossbar.
Cleverley went down under a heavy tackle by Dier, his right ankle caught between the legs of the Spurs defender, and he was carried from the pitch his injured leg in a brace.
Somehow Everton cleared their lines despite being repeatedly bamboozled by the quick feet of Dembele, but then the Belgium playmaker became the second player to be taken away by medics.
Even upon viewing replays it was unclear exactly how Dembele had been hurt, although his right boot appeared to clip his left Achilles after passing the ball in what was a self-inflicted injury.
Dier had escaped a booking for his tackle on Cleverley but referee Mike Jones reached for the yellow card when he body-checked Ross Barkley.
The pressure was building on Everton’s goal and it took some desperate scrambling defence and a skied shot by Chadli to keep the match goalless.
Romelu Lukaku was making his presence felt with a couple of dubious challenges and the visitors could have taken the lead only for Arouna Kone to steer a glancing header wide.
The chances dried up for Spurs in the closing stages, in part thanks to the arrival from the bench of the tenacious Steven Naismith, who brought renewed fight to Everton’s rearguard action.