Another protest was staged outside Ewood Park by disgruntled Blackburn fans prior to kick-off, and the heat was turned up further on Kean when van der Vaart put Spurs in front inside the opening quarter of an hour.
The hosts equalised through Mauro Formica's fierce half-volley in the 28th minute, but van der Vaart produced a brilliant winner from outside the box eight minutes after the restart.
The result leaves Rovers bottom of the Premier League with just five points from nine games, and the calls for owners Venky's to dismiss Kean are only getting louder.
The club's supporters again made their unhappiness known after the final whistle, with a section in the Ronnie Clayton End remaining in their seats, unfurling 'Kean Out' banners and chanting for the Scot to be sacked.
Tottenham, meanwhile, are up to fifth, their unbeaten run extending to six games in all competitions.
Kean made three changes to his side from the 1-1 draw at QPR with Formica, Simon Vukcevic and the fit-again Morten Gamst Pedersen coming into the first XI.
For Spurs, Aaron Lennon, Kyle Walker and Sebastien Bassong were the only survivors from the team that had started Thursday's Europa League victory over Rubin Kazan, with Emmanuel Adebayor, Van der Vaart, Luka Modric, Gareth Bale and Scott Parker - who was named as captain - among those restored to the side.
Rovers skipper Christopher Samba made the first attempt at goal, sending a looping header over the bar, before Pedersen delivered a long-throw into the box which Brad Friedel managed to punch away while back-pedalling and Junior Hoilett drove forward before firing over.
At the other end the visitors were looking threatening, though, and after Jason Lowe and Gael Givet had made important interceptions to avert the danger, Tottenham broke the deadlock.
Walker beat Givet for pace down the right and pulled the ball back for van der Vaart, who slotted it past Paul Robinson.
Spurs continued to press, with a Bale cross diverting off Martin Olsson to bring a save out of Robinson and the Blackburn goalkeeper then getting down to save at the feet of Adebayor.
Rovers appeared vulnerable, but just before the half-hour mark they were suddenly level as Samba's knock-down from Pedersen's free-kick came to Formica, who smashed the ball home.
The goal gave the hosts a lift and Vukcevic saw a strike tipped wide by Friedel before Lowe whipped a ball across the box that Jason Roberts was just unable to get on the end of.
Modric called Robinson into action with a shot from outside the area, but Blackburn finished the first half strongly, with Friedel batting away an effort from Hoilett and the ball going inches wide when Samba and Younes Kaboul rose together to meet Pedersen's free-kick.
At the start of the second half it was Tottenham who began the livelier, Adebayor scuffing an effort high and wide and Robinson parrying a Modric shot, with the follow-up crossed by Lennon but completely missed by Bale.
There was nothing wrong with van der Vaart's accuracy moments later as the Dutchman netted his second of the afternoon, collecting the ball when Benoit Assou-Ekotto's pass ricocheted off Vukcevic and curling it delightfully beyond Robinson.
Blackburn looked to make a swift response and, after Friedel had made a hash of dealing with a long kick direct from Robinson and then a cross from Pedersen, Steven Nzonzi blazed over.
Vukcevic went down in the box under the attentions of Assou-Ekotto but referee Stuart Attwell waved play on, and Olsson put a cross into the danger zone which nobody was able to finish.
Friedel then touched a Samba header over the bar, but Tottenham went close soon after as Robinson blocked a firm strike from Modric.
Kean made a double substitution in a bid to give fresh impetus to his attack, with Aiyegbeni Yakubu and Ruben Rochina coming on for Formica and Vukcevic.
Yakubu was straight into the thick of the action, narrowly failing to convert a Samba header from close range and then sending a shot wide from Hoilett's cross.
Roberts put a header on to the roof of the net and Rovers knew their luck was out when Yakubu met Olsson's cross in front of goal, only to skew his attempt horribly off-target.