Dries Mertens and Fernando Llorente scored late as Liverpool fell in their Group E game in the San Paolo Stadium on Tuesday night.
The former AFC footballer of the year announced his decision through an Instagram post.
Both Liverpool, who sit top of the Premier League after five straight wins, and Napoli remain favourites to progress from Group E despite Salzburg's thrilling 6-2 win over Genk.
Al Sadd put on a commanding home performance to bounce back from their first-leg loss to Saudi Arabia's Al Nassr and win 3-1 on Monday for a 4-3 aggregate victory.
Barcelona were lucky not to lose after Dortmund captain Marco Reus missed a penalty and several chances in the second half.
Lyon came from behind to draw 1-1 at home to Russian champions Zenit Saint-Petersburg in France.
Hazard was signed for 100 million euros ($110 million) from Chelsea in the summer as the spearhead of a new-look Real, but a thigh injury prevented him from making his first appearance until last weekend.
The Sadd coach picked man-of-the-match Hassan Al Haydos for special praise.
Paris Saint-Germain host Real Madrid on Wednesday to start their Champions League campaign.
It was a poor performance by Vitor Pereira's SIPG side, who were devoid of penetration up front without suspended 55 million euro Brazilian striker Hulk.
Qatar coach Abdul Razzaq Haider has named a 24-member team for the qualifying campaign.
The Cheetahs are unbeaten so far, but have a tough job at hand when they face Al Ahli in the fourth round on Thursday.
The Iraqi striker underwent medical tests, which confirmed a muscle injury on his left foot.
Wearing black hoodies, the pair rushed to a car while refusing to take questions from the media gathered outside the prison.
The world's most expensive player will miss the fixtures against Real Madrid on Wednesday and Galatasaray in two weeks' time.