The holders, Bayern Munich, and Real Madrid showed their Champions League pedigree with convincing wins over Manchester City and Copenhagen.
Franck Ribery, Thomas Mueller and Arjen Robben scored for Bayern in a 3-1 win, while Cristiano Ronaldo found the net twice in Real Madrid's 4-0 drubbing of the Danish side.
Manchester United were forced to settle for a draw when Taison's 76th-minute strike earned Shakhtar Donetsk a 1-1 draw after the visitors had led through Danny Welbeck.
Olympiakos Piraeus romped to a 3-0 win at Anderlecht and, also in Group C, the Sweden striker, Zlatan Ibrahimovic, scored twice as Paris St Germain crushed Benfica by the same scoreline.
A dramatic finale saw Galatasaray snatch a 2-2 draw with Juventus, while Bayer Leverkusen also left it late with a stoppage time goal sealing a 2-1 home victory over Real Sociedad.
CSKA Moscow had a 3-2 home triumph over Viktoria Plzen who have lost their first two games.