Christian Eriksen and Emmanuel Adebayor starred as Tim Sherwood's side overcame a shaky start to produce a thrilling rally, ending former manager Juande Ramos's hopes in the process.
After losing their first leg 1-0 in Ukraine, Spurs fell even further behind when Roman Zozulya added a vital away goal to Dnipro's tally, leaving Sherwood's men in need of three second-half goals.
Spurs duly responded to the challenge as Eriksen netted a free-kick, before Adebayor levelled the aggregate scores by latching onto the end of his Danish team-mate's cross.
Adebayor strucl his second moments later as Tottenham booked a date with Benfica, who beat PAOK 4-0 over two legs.
Tottenham enjoyed the better of the opening exchanges in north London on Thursday night. Roberto Soldado made his way into the Dnipro box early on, yet lacked support as he was crowded out.
Sherwood's side forged the first real opening of the night when Adebayor chested down to Eriksen on the edge of the area, but his strike was deflected wide.
A lively-looking Yevhen Konoplyanka - who was targeted by Liverpool in the January transfer window - tested the Spurs defence, enticing fouls but Hugo Lloris was equal to Dnipro's early set-piece efforts.
Eriksen linked up with Mousa Dembele midway through the first half before striking a long-range effort, which soared high and wide of the target.
Dnipro's best chance of the opening period came when Matheus got the better of Michael Dawson, who appeared to bring the Brazilian striker down, but the referee waved play on.
The Ukrainian side set the hosts a mountain to climb shortly after the restart when Zozulya took advantage of a napping Spurs defence to head home Ruslan Rotan's free kick.
Spurs thought they had equalised when Soldado slotted beyond Denys Boyko after a precision through-ball from Eriksen, only for the offside flag to be raised.
Denmark international Eriksen did restore parity moments later, curling home a superb free kick, which Boyko could only force further into his bottom left-hand corner.
Drama ensued at White Hart Lane as Zozulya was shown a straight red card for an off-ball incident with Jan Vertonghen before the Ukrainian had to be wrestled away by Adebayor; while in the midst of the chaos it appeared Roberto Soldado raised an elbow to Artem Fedetskiy, which went unseen by the officials.
Spurs wasted little time after being handed a personnel advantage, with Eriksen whipping a low cross into the six-yard box, where Adebayor slid in to draw the Premier League side level on aggregate.
Adebayor doubled his tally four minutes later, utilising his chest to control a crossed ball before flicking beyond the keeper to send Spurs into the lead for the first time in the tie.
With the hosts in full flow, Andros Townsend provided Soldado with a fine cross, but the Spaniard was unable to convert as his shot bounced over the crossed.
Despite being reduced to 10 men, Ramos' Dnipro refused to lay down as Konoplyanka drifted into space before unleashing a stinging shot, which forced a diving save from Lloris.
A rash injury-time challenge from Dawson, just outside the Spurs box, gifted Konoplyanka a free-kick in a dangerous position and the most sought-after man in Ukraine stretched Lloris once more with a venomous strike.
Spurs remained resilient however as they saw out five minutes of added time, much to the delight of an overjoyed Sherwood, who raced onto the pitch to celebrate with his side on the full-time whistle.