Match ends, Real Madrid 4, Shakhtar Donetsk 0.
Cristiano Ronaldo moved three clear of Lionel Messi at the top of the Champions League scoring charts after a hat-trick against Shakhtar Donetsk.
His treble, which included two spot-kicks, took him to 80 Champions League goals and eight in his last two games.
Ronaldo's third was a header after Marcelo's shot had been parried.
Karim Benzema had tapped in Real's opener after keeper Andriy Pyatov dropped a cross, while Shakhtar had Taras Stepanenko sent off.
Real Madrid's main cause for concern from the game was forward Gareth Bale having to go off injured after 32 minutes.
The Wales international was holding his calf before being replaced by Mateo Kovacic, while the Spanish club's centre-backs Raphael Varane and Sergio Ramos were also substituted with knocks.
Ronaldo was level with Barcelona rival Messi on 77 Champions League goals before Real Madrid began their quest for a record 11th crown in Europe's top competition.
But after the Portuguese had bagged five goals against Espanyol on Saturday, he scored his third Champions League hat-trick to move clear on his own at the top.
Ronaldo and Messi have constantly overtaken each other for the honour of the competition's most prolific scorer and the latter will have the chance to respond when Barcelona meet Roma on Wednesday.
The first of Ronaldo's spot-kicks came after Darijo Srna was controversially adjudged to have handled the ball, despite a shot from the Real forward appearing to hit him on the back.
"I feel so proud and happy to score eight goals in two games but that is not my goal - my aim is to be good all season and for the team to succeed, which is the most important for me, and win trophies," said Ronaldo.
"It is just the beginning of the season. The league is long. It is too early to say we are amazing or bad. Let's work the same way and let's see what happens at the end of the season."
"Without playing a brilliant match we still created a lot of chances," said Real Madrid manager Rafa Benitez.
"Bale had a spasm in his calf muscle. We must wait a day or two to see how serious it is. He had a similar issue last year."
Elsewhere in the Champions League
Former Real Madrid and Manchester United winger Angel Di Maria scored the first goal of this season's Champions League group stages when he put Paris-St Germain on the way to a 2-0 victory against Malmo.
The Argentine sent an angled shot into the top corner for his first goal since scoring for United in an FA Cup tie against Yeovil in January, with an Edinson Cavani header following Zlatan Ibrahimovic's backheel completing the win.
Wolfsburg brought in Julian Draxler from Schalke as a replacement for Kevin de Bruyne and he scored the only goal in the 1-0 win over CSKA Moscow when he stabbed in a shot after his initial header was saved.
Atletico Madrid made a strong start to their campaign as they won 2-0 in Galatasaray courtesy of two first-half goals from Antoine Griezmann.
Borussia Monchengladbach were back in Europe's top competition after a 37-year absence but lost 3-0 to Sevilla.
The Spanish side had three penalties in the match. Kevin Gameiro scored the first but missed the next before handing over spot-kick duties to Ever Banega, who made no mistake for their second goal.
Yevhen Konoplyanka's shot from the right wing caught Monchengladbach keeper Yann Sommer out at his near post for Sevilla's third.
Champions League debutants Astana were the first team from Kazakhstan playing in the competition but were beaten 2-0 by Benfica after goals from Nicolas Gaitan and Kostas Mitroglou.
- 2VaraneSubstituted forPepeat 45'minutes
- 4RamosSubstituted forNachoat 59'minutes
- 11BaleSubstituted forKovacicat 31'minutes
- 7Cristiano Ronaldo
- 20Rodríguez Ruiz
- 33SrnaBooked at 54mins
- 5KucherBooked at 75mins
- 66de Azevedo
- 6StepanenkoBooked at 50mins
- 11MarlosSubstituted forKovalenkoat 75'minutes
- 28TaisonSubstituted forMalyshevat 67'minutesBooked at 80mins
- 21GladkiySubstituted forBernardat 81'minutes
- 22Alves da Silva
- Ivan Bebek
- Shots on Target
Second Half ends, Real Madrid 4, Shakhtar Donetsk 0.
Foul by Marcelo (Real Madrid).
Darijo Srna (Shakhtar Donetsk) wins a free kick on the right wing.
Corner, Shakhtar Donetsk. Conceded by Keylor Navas.
Attempt saved. Alex Teixeira (Shakhtar Donetsk) right footed shot from outside the box is saved in the top right corner. Assisted by Viktor Kovalenko.
Foul by Nacho (Real Madrid).
Viktor Kovalenko (Shakhtar Donetsk) wins a free kick in the defensive half.
Attempt saved. Mateo Kovacic (Real Madrid) right footed shot from outside the box is saved in the centre of the goal. Assisted by Toni Kroos.
Attempt saved. Karim Benzema (Real Madrid) right footed shot from outside the box is saved in the centre of the goal. Assisted by Isco.
Foul by Isco (Real Madrid).
Darijo Srna (Shakhtar Donetsk) wins a free kick in the defensive half.
Foul by Nacho (Real Madrid).
Fred (Shakhtar Donetsk) wins a free kick in the defensive half.
Attempt blocked. Cristiano Ronaldo (Real Madrid) right footed shot from the centre of the box is blocked. Assisted by Pepe with a cross.
Attempt blocked. Pepe (Real Madrid) header from the left side of the box is blocked. Assisted by Isco with a cross.
Corner, Real Madrid. Conceded by Andriy Pyatov.
Attempt saved. Karim Benzema (Real Madrid) right footed shot from outside the box is saved in the bottom right corner. Assisted by Cristiano Ronaldo with a headed pass.
Substitution, Shakhtar Donetsk. Bernard replaces Olexandr Gladkiy.
Goal! Real Madrid 4, Shakhtar Donetsk 0. Cristiano Ronaldo (Real Madrid) header from the centre of the box to the top right corner following a corner.