Cristiano Ronaldo saw a goal chalked off in Juventus' narrow win at Parma, while Robert Lewandowski scored a hat-trick for Bayern Munich as they hammered Schalke 3-0.
Juventus captain Giorgio Chiellini gave the champions a 1-0 win in the opening match of the new Serie A season, although they were without new coach Maurizio Sarri, who is recovering from pneumonia.
Chiellini flicked the ball home in the 21st minute after Parma failed to properly clear a corner to get the Turin side off to a winning start in their bid for a ninth successive title.
Ronaldo also had the ball in the net before half-time but his effort was ruled out after VAR showed his shoulder was offside by a matter of millimetres when he collected Douglas Costa's pass.
Ronaldo squandered other chances and saw two efforts saved by Parma goalkeeper Luigi Sepe on a frustrating afternoon for the five-time World Player of the Year.
Lewandowski takes home match ball
Lewandowski hit a hat-trick to help champions Bayern Munich cruise to a 3-0 victory at Schalke 04 for their first win in the Bundesliga this season, with Philippe Coutinho making his debut as a second-half substitute.
The 31-year-old Poland forward grabbed his first goal with a 20th-minute penalty and added another with a superbly taken free kick five minutes after the restart.
He completed his ninth Bundesliga hat-trick in the 75th to take his season tally to five goals in two league matches.Bayern, who stumbled to a 2-2 draw in their opener against Hertha Berlin, are on four points, two behind leaders Borussia Dortmund, who won 3-1 at Cologne on Friday.
His second goal looked to be a wake-up call for Schalke who grabbed control of the game and were unlucky not to be awarded a penalty for a handball by Bayern's new signing Ivan Perisic.
Their revival was shortlived, however, and Lewandowski put the game beyond them when he controlled the ball at the edge of the box before completing his hat-trick with a clinical finish.
Real held by Valladolid at Bernabeu
Real Madrid stumbled to a 1-1 draw against Real Valladolid in their first home game of La Liga season after the visitors levelled with an 88th-minute strike from former Barcelona trainee Sergi Guardiola.
Karim Benzema looked to have won the game when he gave the hosts a deserved lead in the 82nd minute with a splendid piece of skill, controlling a pass with his back to goal and swivelling before striking into the far corner.
Luka Jovic had gone close to breaking the deadlock earlier in the game when he headed against the crossbar barely a minute after coming off the bench, while Gareth Bale and James Rodriguez missed chances earlier in the game.
Valladolid had offered little in attack but grabbed a shock equaliser with only their second shot on target when Guardiola struck through the legs of Thibaut Courtois to complete a swift counterattack after Toni Kroos had surrendered possession.
Madrid have four points after two games, having beaten Celta Vigo 3-1 away in their opening fixture last week.