Madrid have won three of their opening six competitive games of the season, including a much-needed 1-0 victory over Sevilla on Sunday that eased the pressure on coach Zinedine Zidane.
With three wins from five in La Liga, Madrid are level with leaders Athletic Bilbao on 11 points, but are trailing because of goal difference.
Hazard, who has made just two league appearances this season after a thigh injury, said there was always pressure to deliver at Madrid and the playmaker is confident of success despite the slow start.
The Belgian joined Madrid from Chelsea in a deal worth €100 million (£88m/$112m), and signed a five-year contract at the Bernabeu.
"When you play for Real Madrid you have to win every game, that's how it is, that's why it's the best club in the world, because you always have that pressure," Hazard said on Monday after being named in the FIFA World XI.
"But we have a quality team. It was difficult but we won at Sevilla, a stadium where it is never easy.
"In the Champions League too, it's still important even if we lost our first game against Paris, but we can dream of a great season with the team we have."
Hazard was speaking of the 3-0 belting by PSG to Madrid in their Champions League opening match in Paris last week.
Despite missing the likes of Neymar, Kylian Mbappe and Edinson Cavani, the Parisians cruised to victory against a lacklustre Los Blancos outfit, with former Madrid star Angel Di Maria proving especially deadly with the first two goals of the match.
Still overwhelmingly favoured to progress from group A, Madrid will host Club Brugge next week as they look to get their European campaign back on track before facing a tough trip to Turkey's Galatasaray on October 22.
But first, Madrid must focus on two La Liga encounters as they first host Osasuna on Wednesday, before visiting Atletico in the derby three days later.