Having lost 1-0 in Dnipropetrovsk last week, Tim Sherwood's side were further up against it when Roman Zozulya headed the visitors ahead shortly after the break before rallying to win 3-2 on aggregate.
But the tie swung back in their favour in 13 second-half minutes as Christian Eriksen levelled on the night before Zozulya received a red card for a headbutt on Jan Vertonghen, who seemed to go down easily.
Spurs soon made light work of turning around the tie as Adebayor struck twice in four minutes to set up a mouth-watering tie with Benfica.
In a week where no Barclays Premier League team had won since both the Champions League and Europa League returned from their winter hiatus, Spurs had struggled to create chances before the interval, with Jermain Defoe's half-time farewell to the club looking ill-timed.
But it appeared to act as a wake-up call to his former team-mates and, once Dnipro had been reduced to 10 men there was only ever one likely winner.
Yevhen Konoplyanka was once more Dnipro's main threat as the match-winner in Ukraine last week tormented Kyle Naughton again.
Roberto Soldado was lucky to remain on the pitch after he caught Artem Fedetskiy with an elbow, having looked around before swinging his arm.
Matheus, who was felled for the late penalty in Ukraine last week, went down inside the box under pressure from Michael Dawson but Gautier rightly waved away his protests.
Sandro fired well over as the first half drew to a close and Spurs continued to toil in their attacking ventures.
The returning Brazilian was then cautioned on the stroke of half-time after a high challenge on Ruslan Rotan.
Defoe appeared at the break for an emotional farewell as Tottenham's exit from Europe also appeared to be closing in as the second half got under way.
Andros Townsend fouled the busy Konoplyanka and Zozulya nodded home the resulting free-kick with Hugo Lloris only able to get a despairing hand to the ball.
Mousa Dembele came close to an equaliser but could only hit the loose ball into the ground and see it bounce onto the roof of Denys Boyko's goal.
Soldado then had the ball in the back of the net moments later but he was adjudged offside from Eriksen's through-ball, but the equalising goal was not long in coming.
Eriksen bent a free-kick out of the reach of Boyko after Adebayor had been fouled on the edge of the Dnipro box and the strike acted as a catalyst to springboard Spurs into the next round.
The hosts, however, were living dangerously and Konoplyanka struck a post with a curling effort, with Fedetskiy firing the rebound high and wide.
Zozulya's moment of madness came just after the hour as the forward headbutted Vertonghen - the merest of touches prompting a theatrical fall - as the Dnipro players formed a wall to defend an impending Spurs set-piece.
A fracas ensued but referee Gautier had seen the indiscretion and dismissed Zozulya without hesitation.
It took less than two minutes for Tottenham to make their one-man advantage count as Adebayor turned Eriksen's low cross past Boyko.
The comeback was complete three minutes later as Adebayor raced clear of the defence before poking his second goal of the evening.
Soldado's hunt for a first goal of 2014 continued as he turned over a difficult chance from close range, with Spurs attempting to kill off the tie.
The expensive summer signing was given a great reception by the home fans when replaced by Harry Kane late on but if it were not for the goals of strike partner Adebayor, Spurs would have been heading out of the Europa League at the hands of former manager Juande Ramos.
Only a late free-kick from Konoplyanka threatened to take Dnipro through on away goals but Spurs saw out five minutes of additional time to seal the win.