Trailing 1-0 from the first leg, two goals from Emmanuel Adebayor saw Tottenham win 3-1 on the night, after Zozulya had scored and then been sent off in the space of 15 second-half minutes.
The Ukraine international looked to have given Spurs a mountain to climb when he headed home at the start of the second half to leave Tim Sherwood's men needing three goals in a little over 40 minutes.
Tim Sherwood named a more attacking line-up than the one that started the first leg in Ukraine, with Adebayor and Eriksen among those 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.
Tottenham held on comfortably in the final stages to set up a final-16 tie with last year's runners-up Benfica, although they will be without their record goalscorer in European competition Jermain Defoe, who bid farewell to the home fans at half-time ahead of his move to MLS outfit Toronto.
