Jose Mourinho's men took the lead through Fernando Torres but goals from Adrian Lopez, Diego Costa and Arda Turan ensured Atletico set up an all-Madrid final in Lisbon next month.
After the match Hazard suggested that chasing the game with Diego Simeone's men defending in numbers was never going to suit his side.
"I don't know what we lacked," he told beIN Sport. "After scoring, we had the game in our hands. Maybe they wanted it more than we did... It was an intense game. Beautiful to watch with opportunities.
"I'm sad but logic has been respected because they deserved to qualify.
"Chelsea is not made to play football. We're good on the counter, a little bit like Real against Bayern.
"We did it well tonight since we scored. But their goal hurt us before the break. We failed to make the difference in the second half.
"Often, I'm asked to do it all by myself and it's not easy. It was a complicated game. It will be a lesson for us regarding next year."
Hazard stoked the fire of speculation by donning a PSG shirt for an interview with Canal+ after Chelsea's 3-1 defeat to Laurent Blanc's men at the Parc des Princes earlier this month.
The 23-year-old also admitted in an episode of French documentary series Interieur Sport that he would have to consider a move to the French capital if his wife insisted on it.