The Serie A champions came from behind to seal a 2-1 victory at Wembley on Wednesday, moving into the quarter-finals 4-3 on aggregate.
Argentina internationals Gonzalo Higuain and Paulo Dybala struck in the space of three second-half minutes to turn the tie around.
And Allegri said Juventus timed their goals perfectly as last season's runners-up moved into the last eight of the Champions League.
"The lads and club deserve compliments for what they’ve been doing over the last few years," Allegri told Mediaset Premium.
"It was a prestigious victory at a stadium like Wembley.
"We waited for the right time, showed patience and won the game. Over the two legs, I think our qualification was deserved.
"We struggled in the first half, above all by losing the ball, and they were deadly in those situations.
"We kept splitting the team in two with misplaced passes, but after the break were more balanced, compact and kept it tight.
"The lads achieved something impressive."
Higuain returned from two games out with an ankle knock, while Dybala started after missing the first leg with a hamstring problem.
But although the duo came up trumps with the goals, Allegri believes introducing Kwadwo Asamoah and Stephan Lichtsteiner shortly before the strikes was a pivotal moment.
"I made changes because we needed players with different characteristics, as Asamoah is extraordinary in his passing and has been really underestimated throughout his career," Allegri added.
"Lichtsteiner did well working with [Sami] Khedira. We were tired on the flanks and needed some more pace and width. Tottenham hit the post in the final minute, but Harry Kane was offside.
"Higuain had been out for two weeks, Dybala only really returned on Saturday, so we need to get the others back to full fitness. Now we have a couple of games coming up in Serie A that will be crucial."