Spurs goalkeeper Hugo Lloris chose not to dive in either direction as Juventus forward Gonzalo Higuain approached to strike his second penalty of the game. With no big gaps appearing, Higuain smashed the ball straight down the middle and saw it cannon off the crossbar.
A goal would have given Juventus a commanding 3-1 lead at halftime. Instead, Spurs managed to recover from a terrible start and level in the second period to finish at 2-2, ahead of the second leg at Wembley on March 7.
Lloris had dived low to his right against Higuain's successful first spot kick, getting a hand to the well-placed penalty.
Harry Kane and Christian Eriksen scored in each half after Higuain had netted twice inside the opening nine minutes
"I feel so proud," Tottenham coach Mauricio Pochettino said. "We showed great character against a team that played two finals in three years, with amazing experience on the pitch.
"We are a young team in this competition. To show that performance is to be proud and to congratulate the players because they were fantastic."
Juventus coach Massimiliano Allegri reacted strongly afterward when it was suggested his team, which is going for a seventh successive Serie A title, should have beaten Spurs, which is fifth in the Premier League.
"I think people have lost a bit of objectivity," said a clearly annoyed Allegri. "Juve plays to win but it's not favorite. You can't go to the final every year. I think the lads are having a great season, but it's a long leap from there to think they're going to win a last-16 match 3-0.
"That drives me nuts, because people don't have an idea about the other teams. I can't accept that people are getting depressed over a 2-2 draw, it seems like we lost 8-0."
It was also an exciting first half in Basel, with Manchester City scoring three goals before going on to win 4-0 to leave it with one foot in the quarterfinals.
Allegri went for an attacking lineup, with Federico Bernardeschi, Douglas Costa and Mario Mandzukic supporting Higuain.
And that paid off right from the start in a terrible opening 10 minutes for Spurs.
Mousa Dembele fouled Miralem Pjanic, 30 yards (meters) from goal, inside the first minute. Pjanic himself took the free kick and picked out Higuain, who beat the offside trap to volley into the far corner.
Matters swiftly went from bad to worse for Spurs as Ben Davies tripped Bernardeschi inside the area and referee Felix Brych pointed instantly to the spot.
Higuain stepped up and placed a precise penalty into the bottom left corner for his second goal of the match and his eighth in his last five matches.
Tottenham had a mountain to climb against a side which had reached the Champions League final in two of the past three seasons. Juventus had not let in a goal in 2018 and had only conceded one in its past 16 matches.
However, with Juventus content to sit back, Spurs began to dominate possession and almost got one back in the 26th but goalkeeper Gianluigi Buffon pulled off a great save to keep out Kane's header from point-blank range.
Higuain almost got his hat trick after a swift counterattack and some lovely interplay with Pjanic, but he fired narrowly wide of the left upright.
Buffon did well to tip another Kane effort round the post but Spurs were getting closer and did get back into the match 10 minutes from halftime.
Kane ran onto Dele Alli's through ball, took it round the onrushing Buffon and calmly deposited into an empty net.
It was the first goal Juventus had conceded in 696 minutes
Spurs pushed for the equalizer but Juventus almost extended its lead on the stroke of halftime when Serge Aurier brought down Douglas Costa in the area. Higuain stepped up again but failed to complete his hat trick, firing onto the crossbar.
He had another good chance to grab a third, shortly before the hour, but he headed a corner straight at Lloris, who had done well to tip Bernardeschi's effort round his post moments earlier.
Juventus was made to pay for its missed opportunities as Buffon was caught out by a low Eriksen free kick into the bottom left corner.
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