Piquionne's first-half header also gave West Ham their first league win of the season in a thrilling London derby at Upton Park.
It was no more than West Ham deserved as they showed the grit and determination which manager Avram Grant will need to see more of if they are to avoid relegation this season.
Goalkeeper Robert Green also made a string of athletic saves as he continued his rehabilitation following his World Cup howler for England, including one from Luka Modric in the first half which was a candidate for save of the season.
West Ham were hoping to build on their midweek Carling Cup victory against Sunderland and the confidence oozed from them, especially in the first half.
They might have taken the lead inside the first minute when Victor Obinna dragged a left-foot shot wide when clean through, but that set the tone for a match which swung from end to end.
A Kieron Dyer shot deflected just wide and a Manuel da Costa header also went close before the Hammers received their reward for their enterprising play in the 29th minute.
When it came, however, it had little to do with thrills. It was simple and direct, Mark Noble swinging across a swerving corner for Piquionne who rose above the rest to send a glancing header into the Tottenham net.
It was no more than West Ham deserved as they were playing football which belied their position in the relegation zone, one mazy Scott Parker dribble into the area typical of their increasing confidence.
But if the Hammers attacking play was impressive then it would have counted for nothing without Green.
Green has taken some fearful stick since his howler against the USA at the World Cup, but there is no doubt that on his day he is a supreme shot-stopper.
He needed to be, tipping away a thunderous drive from Jermaine Jenas and palming out a 20-yard left-foot shot from Van der Vaart. But it was his save to deny Luka Modric which was world class.
Brilliant work by Aaron Lennon down the right saw the ball arrive for Modric to swing his cultured left boot. It was the sweetest of strikes, which looked goalbound until somehow Green threw himself to his right to tip it onto the bar and away to safety.
Tottenham had their chances in the second half, Alan Hutton putting Tom Huddlestone clear with just Green to beat. The Tottenham captain did the hard bit, rounding the goalkeeper, but then contrived to send his tame shot horribly wide of the post.
Carlton Cole came on for the limping Piquionne but it was West Ham who should have extended their lead when Obinna got the better of the Spurs defence and fired in a left-foot shot which was brilliantly saved low down by Carlo Cudicini.
The Tottenham goalkeeper also denied Noble in the 73rd minute when he somehow managed to turn the midfielder's goalbound drive acrobatically around the post.
Redknapp threw on Robbie Keane for the ineffective Lennon in a bid to supply more fire power but Keane could only shoot straight into the hands of Green when his clearest chance presented itself.
One sour note for the heavily-criticised Grant was seeing an injured Dyer leave the field after 41 minutes to be replaced by Pablo Barrera. But it was a match from which he will take great heart.