Second-half goals from Heung-Min Son, Jan Vertonghen and Fernando Llorente left Spurs in sight of the last eight.
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The home side had the first chance of the night, Lucas Moura flashing wide an improvised half-volley, but in the main the opening half belonged to Dortmund.
Hugo Lloris made fine saves from Christian Pulišić and, on the stroke of half-time, Dan-Axel Zagadou to keep Tottenham level – and yet two minutes after the restart Spurs struck, the in-form Heung-Min Son volleying in Jan Vertonghen's pinpoint cross.
Tottenham looked in control thereafter and, with Dortmund running out of ideas, Vertonghen crept in to smash in a late second from Serge Aurier's cross before Fernando Llorente nodded in Christian Eriksen's left-wing corner.
View from the stadium: Daniel Thacker
Tottenham were patient and got their reward. After withstanding Dortmund's first-half pressure, Spurs were clinical after the break, Son, Vertonghen and Llorente finishing their chances expertly for the club's first ever home win in a UEFA Champions League knockout tie.
Buoyed by a three-goal advantage and with Harry Kane and Dele Alli potentially available for the return, Mauricio Pochettino's troops will expect to complete the job in Germany.
Man of the match: Jan Vertonghen (Tottenham)
The Belgian was outstanding in an unfamiliar role at left wing-back, defending stoutly in the first half to snuff out the threat from Jadon Sancho and Achraf Hakimi. In the second period he was a man transformed, rampaging down the flank to devastating effect. His cross for Son's opener was sumptuous, bettered only by the expert finish for Tottenham's second goal.
- Spurs have won a home #UCL knockout match for the first time (D1 L2); they had scored only one goal before tonight
- The Lilywhites are well set to reach the #UCL quarter-finals for just the second time, after 2010/11
- Son has scored in his last four games against Dortmund, and has nine goals in his 11 appearances versus BVB
What does it mean?
Tottenham have lost only once in their last five visits to Germany, although that defeat came at Dortmund in the 2015/16 UEFA Europa League round of 16 first leg (0-3).
Moreover, the London club have only twice lost a UEFA-competition tie after having won the home first leg, and both of those home victories had been by a single-goal margin. Dortmund, meanwhile, have won nine of the 16 ties in which they lost away in the first game and will have it all to do on 5 March.
Spurs now have an extended rest period before Premier League away games at Burnley (23/02) and Chelsea (27/02). Then, just three days before the second leg, they welcome north London rivals Arsenal – currently ten points adrift of them – to Wembley.
Top play bottom when Dortmund visit Nürnberg on Monday, with a home match against Leverkusen to follow on 24 February. Lucien Favre's side then go to Augsburg on 1 March.