Trailing 2-0 from the first leg, Dnipro's hopes of turning the tie round looked to have disappeared when substitute Nikola Kalinic was sent off in the 63rd minute - just 10 minutes after climbing off the bench.
However, the hosts were handed a lifeline 14 minutes from time when Basel midfielder Fabian Frei handled inside the area, earning him an early bath and the home side a penalty.
Yevhen Seleznyov duly slotted home from the spot to put Dnipro ahead but their hopes of staging a late comeback were crushed just five minutes later when Basel's 10 men won a penalty of their own, which Fabian Schar converted.
That away goal ended Dnipro's challenge, as well as their coach Juande Ramos' chances of this season becoming only the second manager to win the competition three times following his successes with Sevilla.
Although Basel held a healthy lead, Dnipro would have felt reasonably optimistic about turning things around having won their last five European home matches, included impressive victories over PSV Eindhoven, Napoli and 2010 Europa League finalists Fulham.
They were also the top scorers in the group stages with 16 goals.
However, they had never won a European match in the new year during what is traditionally their mid-winter break from domestic action, and that proved to be the case again on Thursday night.
Ramos' men, whose only competitive fixture in the last two months was the first leg against Basel, bossed the first half but the closest they came to a breakthrough was Yevgen Konoplianka's brilliant 33rd-minute free-kick which was parried away by Yann Sommer.
They also started the second half on top but suffered a huge blow when replacement Kalinic was shown a straight red card for his foul on Serey Die.
The Ukrainian outfit refused to give in though, and they came close to scoring through Konoplianka before taking the lead when Seleznyov fired home from the spot after Frei was sent off for deliberate handball.
That should have set up a grandstand finish at the Dnipro Arena, but just five minutes later the hosts conceded a crucial away goal.
Roman Zozulya's foul on Valentin Stocker saw Basel awarded a penalty and Schar smashed home from 12 yards.