The Hammers were 3-0 up within 10 minutes, with Vaz Te grabbing two before getting his third in the second half, and Allardyce was delighted with his side's performance, reserving special praise for the Portuguese forward.
"It was a terrific performance. We could arguably have scored more than three in the first 10 minutes. I'm pleased the lads never gave up, came out second half and got control again. We've got within three on goal difference of Southampton which hopefully could put some more pressure on them," said Allardyce.
"The West Ham fans saw the team playing at home like they play away from home. We got six goals and the all-important clean sheet.
"We picked up the golden nugget with Ricardo Vaz Te. He's always had the ability since I had him at Bolton and has grown up at last and got the bit between his teeth. The first goal was sheer power with dip swerve and movement and the third was the icing on the cake for him."
The Hammers now sit just three points behind second-placed Southampton in the Championship table, with three games left in the race for automatic promotion.