Bale opened the scoring in the first half and struck a long-range winner in the 90th minute to hand Spurs all three points.
Allardyce praised the Welshman's recent form but admitted that he was disappointed at his side not taking their chances to win the match.
"We've one man to thank for tonight's result and that's Gareth Bale. He's the difference and his wonder goals are continuing," said Allardyce.
"I can't say I've ever seen anyone score so many in such a short time. He's done it twice tonight.
"The first was a good goal but the second in injury time is a a wonder goal. It's a hugely disappointing result after getting back in the game and taking the lead. You have to take your chances when they come and Matt Taylor had the chance but he didn't take it."
The result lifts Tottenham up to third in the Premier League table, two points clear of fourth-placed Chelsea.