The Togolese striker netted twice in four second-half minutes to turn the tie on its head after Christian Eriksen's free kick had canceled out Roman Zozulya's opener.
Roman Zozulya headed in Yevhen Konoplyanka's free kick on 47 minutes to put the Ukrainian giants two goals in front on aggregate, but Christian Eriksen's free kick pulled the hosts level on the night.
Adebayor struck twice in four minutes after Zozulya had seen red for headbutting Jan Vertonghen to turn the tie on its head and fire Tim Sherwood's men through.
Sherwood named a more attacking lineup than the one that started the first leg in Ukraine, with Adebayor and Eriksen among the big names to return.
But despite the more adventurous selection, it was 25 minutes until the hosts managed a meaningful effort on goal - Eriksen's 30-yard free kick comfortably saved by Denis Boyko low down to his right.
After a first half devoid of action, the hosts suffered a nightmare start to the second period, Zozulya stealing in unmarked to head home Ruslan Rotan's whipped free kick from the left.
Rather than deflate them, the goal appeared to spark Spurs into life, Mousa Dembele thumping a shot into the turf and onto the roof of the net, before Roberto Soldado's first-time effort was denied by an offisde flag.
Eriksen did pull one back in the 56th minute, sending a dipping free kick around the wall and into the bottom left-hand corner, despite Boyko's strong hand.
The second half was unrecognizable from the turgid fare on offer in the first, former Tottenham target Yevhen Konoplyanka curling a low effort against the post from 25 yards with Hugo Lloris beaten.
Shortly after the hour, Zozulya put his side under unnecessary pressure by headbutting Vertonghen as Dnipro prepared to defend another Eriksen set piece.
Ten minutes later, two goals from Adebayor turned the tie completely on its head.
Eriksen was involved yet again for the first, sending in a devilish low ball for the Togolese, who sprung the offside trap and tapped home from six yards.
Four minutes later, Spurs were ahead for the first time in the tie, Ezekiel Fryers sending a long ball forward which Adebayor controlled beautifully on his chest before poking the ball beyond Boyko.
Tottenham held on comfortably in the final stages to set up a last 16 tie with last year's runner-up Benfica, although it will be without its record goalscorer in European competition. Jermain Defoe, who bid farewell to the home fans at halftime ahead of his move to MLS outfit Toronto.