There were few chances in a match in which United full-back Rafael was sent off in the second half for a second bookable offence.
But Redknapp believed his side had the best of the chances, with strikes from Rafael van der Vaart and Peter Crouch, and insisted Tottenham were getting closer to matching the Premier League big boys.
No way past: Spurs matched but couldn't kill off United at White Hart Lane
Redknapp, whose side remain fifth, one point behind Chelsea in the race for the Champions League spots, said: 'What it does is show we are not a million miles behind the other teams. We played United close, we battered Man City here first game of the season and drew.
'We have to believe we are not inferior to anyone. We are getting closer and closer. The top five teams are getting closer and closer. We are outsiders, but the favourites will be Man United still.'
Redknapp, however, does not believe United will go through the season unbeaten and emulate Arsene Wenger's famous Arsenal 'Invincibles'.
Leap of faith: Redknapp says Spurs are close to matching the top flight's big boys
Redknapp said: 'I couldn't say they (United) are on another level. I can't see them going the year unbeaten. There are lots of teams who could beat them on any given day.
'You can't be as good a team as you were a year or two ago when you had (Cristiano) Ronaldo and (Carlos) Tevez. I think other teams have improved and got closer to them.'
Redknapp had some sympathy with United manager Sir Alex Ferguson over the sending off of Rafael for a foul on Benoit Assou-Ekotto. He said: 'He's just run in and clipped his (Assou-Ekotto) heels. He hasn't stuck his leg out. It was Mary Decker and Zola Budd wasn't it? What can you do about that? He hasn't broken his stride.'
Time running out: a loan move for David Beckham is looking increasingly unlikely, says Redknapp, with the player due back at the LA Galaxy in February
Redknapp also admitted time was running out in his bid to sign David Beckham on loan. He added: 'David has to be back there (America) in February and there's not a lot of time, which is a shame.'
Tottenham are also unlikely to sign Everton's Steven Pienaar after the Merseyside club accepted a £3million offer from Chelsea.
Redknapp also appeared to rule out any chance of landing Newcastle's Andy Carroll.
'He's a good player, he's a handful, you'd like to have him in your team,' Redknapp said. 'But we are nowhere near getting Andy Carroll. How much is he? You are talking £30million to 40million.'
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