Shakhtar overlooked a series of chances of winning by an even bigger margin as they dominated throughout the match with crisp one-touch passing and superior close control.
Shakhtar won second-tier European competition in 2009.
After Djordje Nikolic rushed off his line to meet a Marlos corner he was beaten to the ball by the striker, later Moraes headed the Ukrainian side into a second-minute lead following a hurler from Basel’s Serbian goalkeeper Nikolic.
Nikolic tried to make amends in the first half with several big saves but was powerless in the 22nd minute when Taison capped a lightning Shakhtar break with a deflected shot after good work by Marlos and Patrick’s clever dummy.
Before Patrick made it 3-0 with a 75th-minute penalty after Yannick Marchand troubled Taison in the area, Marcos Antonio jolted the woodwork, with a long-range effort in the 40th minute.
In the 89th minute with an irresistible shot right-back Dodo put the icing on Shakhtar’s pie before Ricky van Wolfswinkel, Basel’s replacement, netted the Swiss from near range a stop-time concession.
Shakhtar Donetsk – FC Basel 4-1
Goals: Junior Moraes 2, Taison 22, Alan Patrick 75, Dodo 88 – van Wolfswinkel 90+2.
Yellow Cards: Alan Patrick 52, Bondar, 87 – Frei 26, Cabral 64.
Shakhtar Donetsk: Pyatov, Dodo, Krivtsov, Bondar, Matvyenko, Marcos Antonio (Maycon 85), Stepanenko, Marlos (Solomon 72), Alan Patrik (Kovalenko 77), Taison (Tete 86), Junior Moraes (Fernando 85).
Basel: Nikolic, Widmer, van der Werff (Rodrigues 73), Alderete, Petretta, Xhaka (Marchand 60), Stocker (van Wolfswinkel 73), Campo, Frei, Pululu, Cabral (Ademi 73).