Cristiano Ronaldo hit a second-half hat-trick as Real Madrid brushed aside 10-man Shakthar Donetsk 4-0 in Group A at the Santiago Bernabeu.
Karim Benzema put Real ahead in the 30th minute and Ronaldo made Shakhtar pay for the dismissal of midfielder Taras Stepanenko for a second yellow card five minutes after the re-start.
Ronaldo dispatched two penalties in eight second-half minutes before grabbing his third goal nine minutes from time as Real made light of the first-half loss of Wales forward Gareth Bale with a calf injury.
The treble meant the Portuguese star moved three goals ahead of Lionel Messi to sit alone at the top of the all-time Champions League scorer's charts on 79.
Former Real star Angel Di Maria was also on target in Group A as Paris St Germain beat Malmo 2-0 at home.
The former Manchester United winger's last club goal came at Yeovil in an FA Cup tie in January but he ended an eight-month wait to find the net four minutes in and Edinson Cavani sewed up victory just after the hour mark.
Antoine Griezmann struck twice in the first half as Atletico Madrid made an ideal start to their Group C campaign with a 2-0 victory at Galatasaray.
The France forward fired Atletico in front after 18 minutes and he doubled their lead seven minutes later by poking home Diego Godin's headed knock down.
Benfica opened their Group C campaign with a 2-0 win over Kazakh new boys Astana in Lisbon.
Astana - a club only founded six years ago - became the first Kazakh side to play in the group stages of the Champions League but second-half goals from Nicolas Gaitan (51) and Kostas Mitroglou (62), the Greek striker on loan from Fulham, gave Benfica victory.
Manchester United fell to a surprise 2-1 defeat to PSV Eindhoven in a Group B game overshadowed by a terrible first-half injury to Luke Shaw, who broke his leg in two places.
United's next opponents Wolfsburg - the German side visit Old Trafford on 30 September - will head to Manchester with three points already in the bag after their 1-0 home victory over CSKA Moscow.
Julian Draxler, signed as a replacement for Manchester City's new signing Kevin De Bruyne last month, settled the contest five minutes from half-time after CSKA Moscow goalkeeper Igor Akinfeev had saved his initial effort.
Manchester City fared little better than their neighbours as they surrendered a lead to lose their Champions League Group D opener 2-1 to Juventus, the Italian champions' first win in England since 1996. An own goal by Chiellini gave City the lead but goals from Mandzukic and Morata turned the tables.
City's next Champions League test in two weeks' time is away to Borussia Monchengladbach and the German side will be seeking to recover from a 3-0 defeat at Sevilla.
The Spaniards were awarded three second-half penalties, converting two of them, after a goalless first period.
Kevin Gameiro scored one and missed one before earning another spot-kick which was converted by Ever Banega. Ukraine winger Yevhen Konoplyanka wrapped up the victory six minutes from time with his first goal for the club.