The 20-year-old beat Bayern Munich winger Jamal Musiala and RB Leipzig's Xavi Simons to pick up the award that recognises players under the age of 21.
The England midfielder, who moved to Madrid from Borussia Dortmund in June, netted his sixth goal in six games this season.
The 20-year-old once again led the Spanish giants' attack, scoring for the fourth time in his first three league games following Rodrygo's missed penalty earlier in the second half.
Player-of-the-match Bellingham, signed for 103 million euros plus 30 million euros in add-ons from Borussia Dortmund, looked unstoppable as he made relentless challenges and created several opportunities up front.
Marco Rose says he will do everything to keep Bellingham and Haaland at the Bundesliga club as the young stars attract the attention of Europe's top clubs.
Southgate said he could hand Borussia Dortmund teenager Jude Bellingham his first cap when they host Ireland on Thursday.