Harry Kane fired home in the 35th minute to halve the deficit – the first goal Juventus have conceded in 2018 – before Higuain wasted the chance to score a hat-trick when his penalty on the stroke of halftime hit the crossbar.
Tottenham were rewarded for their endeavour when Christian Eriksen’s free kick levelled it up in the 71st minute, meaning the Londoners will fancy their chances of knocking out last year’s runners-up in next month’s second leg at Wembley.
Tottenham arrived full of confidence after winning their group containing last year’s winners Real Madrid, but they suffered a dreadful opening 10 minutes.
Juventus, unbeaten in their previous 26 home matches in Europe, took the lead after two minutes when a clever free kick by Miralem Pjanic picked out Higuain who swivelled to volley right-footed past Hugo Lloris.
Tottenham had barely put two passes together when they fell further behind seven minutes later from the penalty spot.
Ben Davies was caught napping and his clumsy foul on Federico Bernardeschi gave the referee an easy decision to point to the spot. Higuain duly converted despite Lloris guessing the right way and getting a touch on the ball.
The rest of the half belonged to the visitors as they shrugged of the double blow to dominate possession.
With Christian Eriksen and Mousa Dembele pulling the strings for Tottenham in midfield, Juve retreated and they were indebted to veteran keeper Gianluigi Buffon for two fine saves to deny in-form striker Kane.
There was nothing he could do though when Dele Alli picked out Kane in the 35th minute and the England striker drifted past the keeper before coolly sliding home.
Tottenham had a huge let-off on the stroke of halftime when they were caught on the counter-attack and Douglas Costa enticed Serge Aurier into a desperate lunge to concede another penalty.
Higuain took the ball but rather than celebrate a first-half hat-trick his penalty rattled against the crossbar.
Juventus played more like the away side in the second half with Tottenham again pressing, although the hosts went close with Lloris saving well from Bernardeschi and Mario Mandzukic then heading straight at the French keeper.
Eriksen, arguably the best player on the pitch, levelled for Tottenham in the 71st minute when his low free kick deceived Buffon and nestled in the corner.