Despite concerns at both ends of the pitch, Carlo Ancelotti's side earned a convincing victory at the San Mames to kick off their title bid, with Rodrygo opening the scoring.
Madrid, who lost Ballon d'Or holder Karim Benzema to big-money Saudi football in the summer, are still heavily linked to Paris Saint-Germain forward Kylian Mbappe as a replacement, as the transfer saga drags on.
Brazilian duo Vinicius Junior and Rodrygo worked well as a strike duo, with Bellingham in the No 10 role at the tip of a diamond midfield offering ample support.
"It's really special, the most important thing is the win," Bellingham told Real Madrid TV.
Los Blancos goalkeeper Andriy Lunin started in place of regular stopper Thibaut Courtois, who is set to miss the majority of the season with a cruciate ligament injury.
The Ukrainian was solid, although he may soon be behind Chelsea goalkeeper Kepa Arrizabalaga in the pecking order, with the Spaniard set to join on loan according to various reports.
Ancelotti, who will take over the Brazilian national team at the end of this campaign, handed defender Fran Garcia his second Madrid debut after his return from Rayo Vallecano.
However it was Bellingham's name that stood out and he demonstrated why Madrid paid over 100 million euros (£88.5m) for him.
"I'm very happy to play with (Bellingham), he's a great player, and I'm sure he'll help us a lot this season," said Rodrygo.
Ancelotti also hailed the England international on his fine debut.
"(He was) very good, he's got personality, being here motivates him a lot," said Madrid's coach.
"He's a fantastic player, a very important signing - he's out of the ordinary."
