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,” he added. “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 said. “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. (AFP)