Pochettino praised his team for their response to falling 2-0 behind inside 10 minutes away from home.
Ilkay Gundogan, Bernardo Silva and Sergio Aguero scored within nine minutes of each other in the last 16 first leg tie to nearly certainly extend Basel's wait for a first quarter-final appearance in the competition.
Juve will travel to London for the second leg on March 7, where Allegri claimed Spurs will be under more pressure to go through.
Spurs remain unbeaten since 16 December, and Pochettino said: "We are more mature". The team is growing and have stepped up in this aspect.
Eriksen admitted: "I think the most important time of the game was the missed penalty which gave us a lot of belief and confidence going into the second half that we could get something and luckily we did".
Yet this performance, on top of wins over Real Madrid and Borussia Dortmund in the group stage, provided further evidence that Spurs are continuing to grow in this competition with each game.
Miralem Pjanic won a free-kick against Mousa Dembele and clipped the ball cleverly into the path of Higuain, who watched the ball drop over his shoulder before volleying across Hugo Lloris' goal and into the bottom-left corner.
Ben Davies was caught napping and his clumsy foul on Federico Bernardeschi gave the referee an easy decision to point to the penalty kick spot.
Gonzalo Higuain struck for Juve after two minutes and again from the penalty spot seven minutes later as Spurs were rocked at the Juventus Stadium, but the Premier League side fought their way back into the match to take a slight advantage going into the return leg.
First Kane had a header saved from inside the six-yard box by Buffon, but the England striker was not to be denied for long, beating the veteran with a composed finish after Alli's pass.
However, with Juventus content to sit back, Spurs began to dominate possession and nearly got one back in the 26th but Buffon pulled off a great save to keep out Kane's header from point-blank range.
The Argentine's penalty kick on the stroke of halftime hit the crossbar.
Lloris produced a fine save from Bernardeschi after the interval but Tottenham got the reward their hard-work deserved when Eriksen scored from a set-piece.
"We're more experienced that past year, we've just got to see how far we can go and take each game as it comes".
"In any football match, you can have ups and downs". All the types we have in our team are players who like to go forward, try to create something. But - and stick with this - their 2-2 draw in Turin, particularly the circumstances in which it was achieved, might have been more valuable than a victory. But I want to congratulate the players.
"The tie is open".
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