Mauricio Pochettino’s side had been under the cosh as Dortmund dominated the first half at Wembley, but Hugo Lloris kept them level with a series of fine saves.
That set the stage for a sublime second half from Tottenham, who were sparked into life by a fine finish from South Korea forward Son.
Son’s fourth goal in his last four appearances took him to 16 for the season, but none have been as priceless as this one.
Tottenham defender Jan Vertonghen capped his own powerful performance when he swept in their second goal in the closing stages and substitute Fernando Llorente’s late third put the seal on a remarkable result.
Although Tottenham still have to survive the second leg at Signal Iduna Park on March 5, they have a golden opportunity to reach the Champions League quarter-finals for the first time since 2011.
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It was an especially impressive success for Pochettino given he was without the injured Harry Kane, Dele Alli, Danny Rose and Ben Davies.
Pochettino devised a clever game-plan that eventually frustrated Dortmund and secured Tottenham’s first ever Champions League knockout-round win at home.
With Rose and Davies sidelined, Pochettino deployed centre-back Vertonghen out of position at left wing-back and switched to a three-man defence that stifled the Bundesliga leaders.
Pochettino had insisted Tottenham would be able to ignore the draining effects of a gruelling schedule which he felt left them at a “massive disadvantage”.
And his passionate talk of seeing the “energy” in his players’ eyes as they looked forward to a big Champions League night proved prescient as they worked tirelessly to leave Dortmund helpless to stem the tide.
Lucas Moura gave Tottenham an early spark when he latched onto Davinson Sanchez’s header and smashed a volley just wide from the edge of the area.
Juan Foyth gifted Dortmund their first sight of goal when he dwelt on the ball too long and was alertly dispossessed by Christian Pulisic.
But Lloris spared Foyth’s blushes as he blocked Pulisic’s shot at the near post.
Dortmund’s rapier attacks were led by Jadon Sancho, who left Manchester City to join Dortmund in 2017 without ever playing for the Premier League team.
Just two years later, the 18-year-old winger made his first appearance back on home soil now established as an England international rated as one of Europe’s hottest prospects.
Sancho hit top gear on several occasions in a mesmerising first-half display, leaving two Tottenham midfielders for dead and linking with Pulisic, whose shot was blocked at the last minute.
It took Tottenham over half an hour before they finally forced ‘keeper Roman Burki into action after Son wriggled through the visitors’ defence.
Lloris made a superb save to deny Dortmund on the stroke of half-time as Dan-Axel Zagadou met Sancho’s cross with a header that the Tottenham captain clawed out at full stretch.
Despite being pushed back for much of the first half, Tottenham had refused to surrender and their perseverance was rewarded in the 47th minute.
When Vertonghen swung over a cross from the left, Dortmund’s defence were out of position for the first time all night and Son timed his run perfectly behind Zagadou to caress a fine volley past Burki.
Suddenly, Tottenham had all the momentum and Vertonghen rampaged forward to fire over.
Dortmund had no answers to Tottenham’s high-tempo pressing and it was Vertonghen who was the hosts’ unlikely hero in the 83rd minute as he volleyed home from Serge Aurier’s cross.
Tottenham were rampant and Llorente completed a memorable night when he glanced a close-range header past Burki from Christian Eriksen’s corner three minutes later