Tottenham Hotspur's Harry Kane and Christian Eriksen scored a goal each to restore the Premier League club's hope of making it past the last 16 of the Champions League.
Juventus had the lead after little more than a minute following a set piece in which Miralem Pjanic sent a pass into Higuaín, who volleyed the ball past Hugo Lloris with a brilliant finish.
And it was soon 2-0 after Ben Davies clattered into Federico Bernardeschi, and Higuan netted his second from the resulting penalty.
Spurs had a penalty shout of their own 10 minutes later when Kane came together with Medhi Benatia, but German referee Felix Brych was not impressed.
Kane then had a chance to get a goal back from six yards in the 26th minute, only to draw a fine save from Gianluigi Buffon.
Spurs were looking more dangerous and the pressure paid off when Kane ran onto Dele Alli’s through ball and rounded Buffon to reduce the deficit.
Just as Tottenham looked to fully bounce back in the game, Serge Aurier mistimed his challenge on Douglas Costa to give Juventus another penalty.
But with a hat trick waiting, Higuain slammed his shot off the middle of the crossbar and the visitors entered the half down only one goal.
Juventus had another chance to restore their two-goal lead in the second half but Lloris made a fine save to deny Bernardeschi, and from the resulting corner Mario Mandzukic sent a header right at the goalkeeper.
And Tottenham made their hosts pay after Giorgio Chiellini fouled Alli in an advanced position, and Eriksen slotted a low free kick into the corner to level the score.
Late on, Benatia put a header over the bar following a corner and Jan Vertonghen did well to cut out a cross, but Spurs were able to see out the respectable result.
In the other game Manchester City trounced Basel 4-0 to secure a comfortable lead into the second leg in England.