Midfielders Oleksiy Gai, Willian and Darijo Srna put the Ukrainians 3-0 up at halftime 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. 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.
Winger Carlitos capitalised on a defensive mix-up early in the Group E match and Federico Almerares struck three minutes from time to wrap up victory for the Swiss.
Galatasaray got off to a flying start 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. Defenders Joseph Yobo and Sylvain Distin got the final touch to corners and Steven Pienaar drove home from 25 metres before Jo sealed the win for the English outfit with a simple finish.
Four times European champions Ajax were held to a goalless draw in Amsterdam by Romania's FC Timisoara in Group A. Celtic paid the price for sitting on the first-half lead Greek striker Georgios Samaras earned them at Hapoel Tel Aviv in Group C. Serb midfielder Nemanja Vucicevic and forward Maaran Lala found the net in the last 15 minutes to earn the Israelis a 2-1 win after Samaras had to come off in the second half having hurt himself falling into one of the dugouts.
Hamburg SV's hopes of reaching the final at their home stadiumwere 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 2-1 victory to Salzburg in Group G through a pair of terrible defensive muddles late on after spurning a series of good chances when 1-0 up.
Genoa stopped it being a washout for the Italians by beating Slavia Prague 2-0 in a Group B match.
