Spurs were cruising after goals by Jermain Defoe and Michael Dawson but Everton hit back through Louis Saha and Tim Cahill before Defoe's penalty was saved by Howard's legs.
The result meant that Spurs remain fourth in the standings on 27, one behind Arsenal after their 2-0 home win over Stoke City on Saturday.
Chelsea, beaten 2-1 by Manchester City, still lead the way on 36 points, two ahead of Manchester United, who won 4-0 at West Ham United.
Spurs were much the better team from the off at Goodison Park but had to wait until the second minute of the second half to score when Defoe turned in a low Aaron Lennon cross.
They doubled the lead on the hour with a diving header by centre back Dawson from a corner.
Injury-hit Everton were without any recognised central defenders after an early injury to Joseph Yobo and Tottenham striker Peter Crouch twice went close to further goals.
Everton had barely threatened but good work by young substitute fullback Seamus Coleman set up fellow sub Saha to pull one back 12 minutes from time and Cahill sent the home fans wild when he stabbed in an 86th-minute equaliser.
Wilson Palacios was then brought down by Tony Hibbert only for Howard to block Defoe's weak spot kick to put the brakes on Everton's slide towards the relegation zone.
Everton boss David Moyes said the late drama gave him a "Big Sam attack" in reference to the heart problems suffered by Blackburn Rovers manager Sam Allardyce last month.
"At 2-0 down against Tottenham with the team we had out I was actually thinking it was going to get worse but credit to the boys, they kept at it and got a great result," he told Sky Sports News.
Spurs manager Harry Redknapp said his side were in "complete control" and that he could not see any danger.
"It was game over at 2-0, they were never going to get back - but we gave a sloppy goal away and we were ruing the opportunities we missed to really kill the game off," he said.
"Then the crowd got behind them and they gave it a real go, they got the second goal - then we had to opportunity to sew the game up but it wasn't to be."
In Sunday's other game Fulham beat Sunderland 1-0 with an early goal by Bobby Zamora, and moved up to eighth.
