Midfielders Oleksiy Gai, Willian and Darijo Srna put the Ukrainians 3-0 up at half-time in one of 24 matches played with additional assistant referees (AAR) beside each goal to spot fouls and dives in the box.
Karel Geraerts pulled one back for the home side just past the hour before Konstantyn Kravchenko's goal wrapped up an emphatic Group J win for Shakhtar in the remodelled UEFA Cup.
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Basel enjoyed a surprise 2-0 win over visiting AS Roma to spoil coach Claudio Ranieri's first European match since he replaced Luciano Spalletti two weeks ago after their poor start in Serie A.
"It was an ugly defeat," former Juventus and Chelsea boss Ranieri told Mediaset television.
"We were not very solid and I saw too few touches of the ball. We have to try to be less beautiful but more concrete. The lads have to learn to react and concentrate for 90 minutes."
FLYING START
Galatasaray, the 2000 UEFA Cup winners, also got under way positively in Group F thanks to a 3-1 win over Panathinaikos in Athens, with the Turkish club's coach Frank Rijkaard getting the better of Henk Ten Cate, once his assistant at Barcelona.
Panathinaikos' city rivals AEK Athens went down heavily too, losing 4-0 at Everton in Group I.
Four times European champions Ajax were held to a goalless draw in Amsterdam by Romania's FC Timisoara in Group A.
Hamburg SV's hopes of reaching the final at their home stadium in May were dented when they were thrashed 3-0 at Rapid Vienna in the other match in the group.
Athletic Bilbao thumped Austria Vienna 3-0 and last year's UEFA Cup finalists Werder Bremen won 3-2 at Nacional Madeira in Group L.
Roma's city rivals Lazio handed a
