The North London outfit found themselves two goals down inside nine minutes but strikes from Harry Kane and Christian Eriksen - either side of a missed penalty from Higuain - gave Spurs a deserved share of the spoils ahead of the second leg at Wembley Stadium in three weeks' time.
Mauricio Pochettino would have instructed his side to be switched on from the opening whistle in Italy given that Juventus were unbeaten in 16 matches in all competitions, a run which had seen them concede just one goal since November 19.
However, within 90 seconds, Miralem Pjanic's free kick caught out the entire Spurs team and it left Gonzalo Higuain in a position to convert a first-time volley into the bottom corner, although replays suggested that the forward was marginally offside.
Spurs were showing intent at the other end but Juventus doubled their advantage from the penalty spot, with Higuain powering the ball past Hugo Lloris after Ben Davies had dived in on Federico Bernardeschi.
The Premier League side were in danger of capitulating but Spurs delivered a defiant response by controlling the remainder of the first half, and Kane should have headed his side back into the game before the half-hour mark, only to be denied by a save at point-blank range by Gianluigi Buffon.
Moments later, Higuain curled wide of the post from just 12 yards - a chance which should have been converted for his hat-trick - but the let-off allowed Spurs to remain in the ascendancy and their pressure paid off with 10 minutes of the first half remaining.
Spurs had worked hard to go into half time with an away goal and just a one-goal deficit, but that nearly changed in added-on time when Serge Aurier lunged in to bring down Douglas Costa inside the penalty area.
However, Higuain opted for power over placement and the striker's effort cannoned off the crossbar and rebounded to safety before the referee blew the whistle for half time.
Spurs had the momentum going into the break but after the restart, Juventus were the better side with their first opportunity through Bernardeschi being tipped wide of the post by Lloris.
From the resulting corner, Mario Mandzukic was left all on his own to get a free header on goal from six yards out but his effort went straight at Lloris.
The same pattern of play continued until the closing 20 minutes as Pochettino kept faith with his starting XI, but Spurs finally began to create some pressure on Buffon's goal.
Kane's long-range effort was bundled away by veteran goalkeeper but the visitors soon won a free kick, which was fired into the bottom corner by Eriksen via the hand of Buffon who was slow to get down to his right.
Pochettino and Spurs were left with the dilemma of whether to stick or twist, but Juventus kept on looking for a third goal and it nearly came when Bernardeschi's deflected effort almost crept past Lloris at his near post.
Mehdi Benatia also headed over the crossbar when he should have at least tested Lloris, but Spurs were able to hang on for what could prove to be an important result in the context of the tie, with the second leg being staged at Wembley on March 7.
JUVENTUS (4-2-3-1) Buffon; De Sciglio, Benatia, Chiellini, Alex Sandro; Khedira (Bentancur 66'), Pjanic; Bernardeschi, Douglas Costa, Mandzukic (Sturaro 76'); Higuain
TOTTENHAM HOTSPUR (4-5-1): Lloris; Aurier, Sanchez, Vertonghen, Davies; Dier, Dembele, Eriksen (Wanyama 92'), Alli (Son 83'), Lamela (Moura 89'); Kane