Despite a 27th-minute straight red card for Jermain Defoe, the Dutch midfielder scored either side of that to take his tally to double figures for the season, leaving Spurs unbeaten in each of the eight games in which he has scored.
Although Marc Albrighton pulled one back eight minutes from time, it was not enough to spare the home side, meaning Redknapp is unbeaten in his last 10 Barclays Premier League games as a manager against Villa.
Van der Vaart, making his return from a hamstring injury, hailed the spirit of his team-mates.
"It was a very good win away with 10 men," Van der Vaart said. "It was difficult, we were down to 10 men, but we did a great job and fought to the end."
Spurs manager Redknapp did not criticise referee Martin Atkinson for Defoe's sending-off, although he was not pleased with the official's overall performance.
"I haven't seen it (the red card) since, but at the time I didn't have any arguments," he said.
"I certainly wouldn't say I think he (Defoe) would go at the player intentionally though. Maybe it was just a little centre forward's challenge.
"The referee is a good referee, but he had a bad game."
He added: "Losing Jermain Defoe obviously made it difficult, but we kept the ball well in the first half. It was a bit more difficult to do that in the second half. I said at half-time we could hit them on the break, which we did."
Villa boss Gerard Houllier was not downbeat despite the loss leaving his side just four points above the drop zone.
"I think it is a poor result, but not a poor performance," the Frenchman said. "Maybe with a bit of luck we could have had a second goal. But their second goal killed us, even if in second half showed we could have got something from the game."