Emmanuel Adebayor struck twice in the first half as Spurs won for the eighth time in nine league matches to move on to 25 points, nine behind leaders Manchester City and four adrift of Manchester United.
Spurs climbed above Newcastle United, also on 25 points, on goal difference and have a game in hand on the teams around them.
Redknapp, 64, back in the dugout after missing two games following minor heart surgery, watched his side dominate the match from start to finish.
"It was great to be back tonight and I felt absolutely fine. I enjoy watching us play football and the best medicine for me is to see the way we played, we dominated the game from start to finish," he told reporters.
"We played some great stuff, had clearcut chances before the first goal went in and in the second half had more chances to kill the game off, but I always thought we were in control.
"Luka Modric and Scott Parker were outstanding in midfield and Younes Kaboul is turning into the player I always thought he could be, his heading was outstanding and we looked assured at the back."
Asked if he thought Spurs had a chance of winning the title, he said he would be happy to finish in the top four and reclaim a Champions League place.
"I don't know about the title, but if we can make the top four again I would be delighted, but it's going to be tight, Chelsea, Arsenal, Liverpool, Newcastle are all doing well and it's going to be tight."
Villa manager Alex McLeish blamed calamitous defending for the goals his side conceded.
"We didn't tackle well, we didn't pass it enough or play well enough," he said.
Adebayor, looking for his first goal in seven games, had a great chance to put Spurs ahead after 11 minutes but headed wide of Shay Given's goal when it looked easier to score, but he made amends three minutes later.
Villa managed to clear a corner but only as far Gareth Bale who sent the ball back into the box where Adebayor acrobatically reacted first to give Spurs the lead with an overhead scissors-kick.
Apart from a first-minute half chance that fell to Emile Heskey and a cross-shot from the former England striker, Spurs totally dominated and it was no real surprise when they doubled their lead.
The Villa defence failed to clear a ball from the left whipped in by Bale and Adebayor stabbed the ball into the empty net after poor defending by James Collins and Richard Dunne.
That goal meant that Spurs had scored at least two in each of their last nine league matches and it could have been worse for Villa at the break with Bale, Kaboul and Kyle Walker all going close.
The second half followed the same pattern with former Spurs striker Darren Bent forcing former Villa keeper Brad Friedel into a good save in the 47th minute before Spurs laid siege to the Villa goal without adding a third.
The result left Villa in eighth place on 15 points, behind the leading quartet and Chelsea, Liverp
Get the best features, fun and footballing frolics straight to your inbox every week.
Thank you for signing up to Four Four Two. You will receive a verification email shortly.
There was a problem. Please refresh the page and try again.