Rafael van der Vaart scored his fifth goal in as many games to send Spurs third in the standings.
The Holland midfielder's effort cancelled out Leighton Baines' brilliant free-kick while Peter Crouch wasted more than one chance to seal what would have been Tottenham's fourth comeback win in five league matches.
"You don't get an easy game with Everton, you never do," Redknapp told Sky Sports. "We had the better of the chances but it was always going to be a tough day on the back of a hard week, but I couldn't fault the effort of the players. They worked hard again today.
"We just needed a break, but it just didn't quite come. But to come back again from 1-0 down shows a lot of character.
"We go behind, we never give up, we come back. That's a fantastic thing in a football team so I'm proud about that."
Redknapp's opposite number David Moyes was equally content with the performance of his players, who have now gone four games without defeat after a slow start to the campaign.
"Tottenham is a really good team at the moment and we're without one or two today, so it was good," Moyes told Sky Sports.
"I thought we did great defensively, the boys stood up to a lot of a balls to [Peter] Crouch. With the pace of [Gareth] Bale and [Aaron] Lennon on either side, they've got a real good balance.
"It was a good point, a hard-earned point. We had a few opportunities on the counter-attack which we didn't take late on, but overall I'm pleased we came here and got a hard-earned point."
Today's stalemate at White Hart Lane comes on the back of a morale-boosting victory over Liverpool in the Merseyside derby, and Moyes hailed his players for their efforts in turning their season around.
"Before Liverpool and Tottenham if you'd given me four points I'd have said I'd take that right away," he said.
"I've got to give the players a lot of credit, they worked really hard for the points. We've come here today without Mikel Arteta, Marouane Fellaini and Jack Rodwell, so the players have done a great job."
Man of the match Baines felt the Everton backline coped admirably in the face of Spurs' myriad attacking options.
"They've got a bit of everything Tottenham and they play some really good football," said the former Wigan full-back.
"They've got some quick players and they can mix it up and knock it long, so it's difficult to know what they're going to do but I thought we fared pretty well against them today.
"It was important that we maintained our unbeaten streak if we can, and a point today is a good result for us."