Despite his costly dismissal against Real Madrid in midweek, Crouch started the game and took only 10 minutes to open the scoring with a well-taken header before Luka Modric made it 2-0.
Crouch added his second after Matthew Etherington had pulled one back for Stoke with a great individual run and finish, and Kenwyne Jones smashed one in off the bar to make it 3-2 just before the break.
Despite an impressive second-half performance, the Potters could not find an equaliser and Spurs held on for a vital three points.
The Londoners will have to put in a much more disciplined performance if they are to overhaul Jose Mourinho's men with five goals on Wednesday, but the win is an important one given that they had failed to win any of their previous four league games in their pursuit of a top-four place.
Just four days after his controversial dismissal in the Bernabeu, Redknapp chose to stick with Crouch up front alongside Roman Pavlyuchenko.
Aaron Lennon failed to overcome the illness that prompted his late withdrawal in Madrid, but Tom Huddlestone and Younes Kaboul made their first appearances in five and four months respectively.
It was a big show of faith from Redknapp to keep Crouch in the starting line-up and the reading of the striker's name over the public address system before the match was greeted with a mixture of cheers and boos from the home support.
Tottenham showed little sign that the 4-0 hammering in Spain had affected them as they started confidently, Gareth Bale drawing an impressive diving save from Asmir Begovic with a 25-yard drive.
The Welshman then crossed for Pavlyuchenko but Robert Huth put in an important block to deny the Russian from eight yards.
Spurs opened the scoring 10 minutes in when Pavlyuchenko whipped in a cross from a short corner and Crouch powered a header past Begovic from eight yards.
The towering hitman, clearly delighted to have scored, ran over to the home fans with his arms aloft to celebrate.
Spurs pushed for a second and it came just seven minutes later. Pavlyuchenko played Modric and the little Croatian breezed past his two markers before slotting the ball between Begovic's legs from for his third of the season.
A Rory Delap long throw drew Spurs goalkeeper Heurelho Gomes into action for the first time, but Stoke were otherwise struggling to get into the game.
However, with 26 minutes gone, the Potters shocked the hosts with a superb goal from former Spurs winger Etherington. He picked the ball up on the halfway line, skipped past Michael Dawson and held off Huddlestone's challenge to go clean through and nutmeg Gomes from eight yards.
Crouch restored the two-goal cushion 33 minutes in when he climbed to reach Huddlestone's pin-point 35-yard cross and power home a well-taken header.
Redknapp was cursing his side's defending soon after, though, as Andy Wilkinson robbed Bale of the ball and played in Jones, who smacked a 20-yard screamer in off the bar.
Pavlyuchenko then wasted two chances and Redknapp cut a disappointed figure as he headed down the tunnel for half-time despite his side being in the lead.
Gomes pulled off a good diving save to deny Jermaine Pennant from a free-kick at the start of the second period as the away side looked for an equaliser.
They almost got it soon after when Jon Walters' close-range effort struck a post after Gomes had palmed away Jones' cross, but Spurs escaped.
A foul on Benoit Assou-Ekotto earned Pennant a booking in the 55th minute before the former Arsenal winger supplied Jones with a beautifully chipped cross but he headed wide.
Stoke were looking dangerous by now and Spurs looked anxious to add a fourth in what was becoming an increasingly open game.
The ball was in the Stoke net 65 minutes in but the referee ruled that Kaboul had bundled Begovic over on the line before Pavlyuchenko tapped in to an empty net.
Begovic then palmed away a fierce drive from Pavlyuchenko before Pennant squeezed a low shot into Gomes' hands after a good Stoke move.
Pulis brought Salif Diao, Ricardo Fuller and Dean Whitehead into the action with 15 minutes left and Jones reminded the hosts of the threat he posed when he fired a 20-yard drive that Gomes did well to tip over.
Redknapp looked to shore up his defence by replacing Assou-Ekotto with William Gallas with three minutes left but despite a nervous period of injury time which saw Gomes punch away two crosses, Spurs held on.
Stoke now turn their attentions to next Sunday's FA Cup semi-final against Bolton and on this evidence can go into the game with confidence.