Hazard was the difference in the pre-season fixture at Salzburg, where he scored his first Madrid goal in sublime fashion on Wednesday.
After arriving from Chelsea in a deal worth €100 million (£91m/$111m), Hazard curled a stunning effort past goalkeeper Cican Stankovic.
"When you're out on the pitch, you want to win the game and being a forward, you want to score goals," said Hazard, who has shown flashes of his brilliance against Bayern Munich, Arsenal, Atletico Madrid, Tottenham and Fenerbahce.
"I'd been waiting for this moment for the last five games and it's happened today. I'm very pleased.
"It's still pre-season, but I think we've got more energy now. We've still got another 10 days before our first competitive game so we have to focus and keep working hard.
"Our aim is to be ready for next week, we can't wait to start La Liga so we're training and practising hard."
Hazard's arrival has headlined a busy off-season for Madrid, who have attempted to overhaul their side with the signings of Luka Jovic, Eder Militao, Rodrygo Goes, Ferland Mendy.
Brazilian defender Eder, who arrived from Porto, made his debut for Madrid against Salzburg. The 21-year-old featured in a back three alongside captain Sergio Ramos and Raphael Varane before making way at half-time.
"It's a special feeling to play my first match for the best club in the world," he said. "I'm very happy, but I still have lots to learn. Playing for Real Madrid has been a dream for me ever since I was very young. Thanks to God, I have the chance to be here and I was to enjoy this opportunity.
"I hope I can have a wonderful career in this shirt. It's an incredible feeling to be alongside the best defenders and fullbacks in world football.
"I hope I can help the team because these are fantastic players. I did what the coach asked of me and I hope things carry on in this vein."