Higuain had put Juventus -- losing finalists twice in the past three seasons -- two goals up within the first nine minutes at the Allianz Stadium.
The Argentine's lightning strike after just 73 seconds was followed by a second from the spot to leave Mauricio Pochettino's English side reeling.
But the 30-year-old missed two further chances to hammer home Juventus's advantage in the first half, with striking the crossbar with a second penalty on the stroke of half-time.
"The team showed real character to come here and challenge a team like Juventus," said Pochettino who blamed the inexperience of his side for their sluggish start.
"I think we weren't focused and it was difficult to manage the first seven minutes. I'm very happy that it's still open and we're going to play the second leg in London."
Juventus coach Massimiliano Allegri warned they would be ready to challenge again for a place in the quarter-finals after extending their unbeaten run at home in Champions League fixtures to 23 games.
"Tottenham had a great game and turned things around.
"After going 2-0 up we stopped playing, but part of that was also down to Tottenham's performance. They are a very physical and technical team."
- Higuain's lightning start -
=============================
Pochettino's side looked down and out after the first ten minutes with Miralem Pjanic lifting a free-kick into the penalty area and an unmarked Higuain volleying past Hugo Lloris, despite being in an offside position.
Higuain's second came from a penalty after Ben Davies fouled Federico Bernardeschi, with Lloris diving the right way but unable to stop the powerful drive.
Buffon twice saved from Kane, including a header from point-blank range, before the striker finally broke through.
Spurs began to take control, only for Serge Aurier to concede another spot-kick by hacking down Douglas Costa, but Higuain was denied a treble by the crossbar.
Kane pulled Spurs back from the brink, latching onto a Dele Alli pass to round a helpless Buffon who had advanced off his line, making no mistake in front of goal.
It was 24-year-old Kane's seventh goal in Europe's top club competition this season and puts him level with Steven Gerrard for the most scored by an English player in a single Champions League campaign.
Asked if he feared his side were about to sink without trace after Higuain's double strike, Kane told BT Sport: "We could have done, away from home in the Champions League. It showed character.
"It was an excellent performance after that and we take two away goals to Wembley. Great result."
Dane Eriksen got the second from a free-kick after centre-back Giorgio Chiellini fouled Alli, sending it low past a wrong-footed Buffon, who should have done better.
"We go into that game full of confidence and we're looking forward to it."
Jan Vertonghen did well to prevent a late Higuain winner when he slid to intercept a Costa cross to prevent the Argentine tapping in.
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