The France striker, last season's European Golden Shoe winner, scored early in the second half after he was fouled in the area by Osasuna defender Juan Cruz.

It was enough to seal the points in Alonso's first game in charge at the Santiago Bernabeu, with Madrid now unbeaten in their opening fixture of a league campaign since 2008.

"You can see that Kylian wants to do even more after this first year where he did great things," said Alonso.

Mbappe finished as La Liga's top scorer with 31 goals in his first season in Spain, and is now wearing the number 10 shirt vacated by Luka Modric after the Croatian left for AC Milan last month.

"I don't know if it's linked to the new number or not, or simply to his desire to win, but he wants to improve individual and collective performance and inspire his team-mates," added Alonso.

"You can see his desire every day. Today he created those decisive moments that enable us to win matches."

Madrid had a shortened pre-season after their run to the Club World Cup semi-finals in July, but Alonso has been adamant that would not be used as an excuse after their request for the Osasuna match to be postponed to a later date was rejected.

"To start on the right foot is key in La Liga. We know we weren't well prepared," Madrid defender Dani Carvajal told DAZN. "But we basically didn't suffer and it was a complete display."

Spain international Carvajal came off the bench for his first league appearance since October after a long-term knee injury, replacing Trent Alexander-Arnold in the second half.

- New faces in Madrid -

Alexander-Arnold made his La Liga debut after his move from Liverpool, with former Bournemouth defender Dean Huijsen and left-back Alvaro Carreras other summer arrivals included in the starting line-up.

Madrid though were without England midfielder Jude Bellingham, who is expected to be sidelined until October as he recovers from a operation to solve a recurrent shoulder issue.

Madrid are looking to turn the page on a disappointing end to the Carlo Ancelotti era after relinquishing both their domestic and European crowns last season.

Alonso's side had to show patience against an Osasuna team that finished ninth last term, with the hosts largely restricted to long-range efforts from centre-backs Huijsen and Eder Militao in the first half.

Osasuna goalkeeper Sergio Herrera was equal to both, while Mbappe failed to connect cleanly after being picked out by Vinicius Junior before he curled another shot wide as Madrid struggled to break down their opponents.

But Mbappe grabbed the only goal six minutes after half-time after he went down following a clumsy challenge by Cruz as he tried to cut past the defender.

Mbappe duly got up and running for the new campaign as he sent Herrera the wrong way from the spot.

Alonso handed 18-year-old Argentine winger Franco Mastantuono his debut midway through the second half as Madrid controlled the game and seldom looked troubled.

Ante Budimir headed over in a rare opportunity for Osasuna and Mastantuono was denied late on by Herrera.

Osasuna finished the match with 10 men with Abel Bretones sent off in stoppage time for throwing an arm to block the run of Gonzalo Garcia.