The Blues responded to the controversial Villa Park defeat with a 2-0 victory over Galatasaray on Tuesday night which secured the 2012 Champions League winners a 3-1 last-16 aggregate win and a place in Friday's quarter-final draw.
Immediate attention returns to the Premier League and Arsene Wenger's 1,000th game as Arsenal boss, against his nemesis Jose Mourinho.
The Frenchman, branded a "specialist in failure" by Mourinho last month, has never beaten the Portuguese in 10 meetings, a streak Hazard hopes will continue at Stamford Bridge.
"The Champions League and the Premier League are both magnificent competitions," Hazard said on chelseafc.com.
"The Champions League is rare because you play less often but the Premier League is so important to us and we know how important it is for the fans too.
"The game against Arsenal comes at an important moment because it follows a league defeat.
"We are going to do our best in front of our own supporters in what is a big derby game. There is no better one to win."
Once again this season, Hazard was the outstanding performer on a night when Stamford Bridge joined in unison to hail Didier Drogba, who returned with Galatasaray after scoring 157 goals in 341 appearances for the Blues, including the decisive penalty in the 2012 European Cup final win in Munich.
Hazard is swiftly inheriting the title of Chelsea's attacking talisman which once belonged to Drogba, who is now 36.
"As I said to him after the game finished, it was a pleasure to play against him," the Belgium playmaker added.
"He is still one of the best strikers in the world. It is true that he is no longer young but that's how we young players learn - by meeting people like him.
"I have nothing but respect for what he has done and what he is still doing. I admire Didier for the player he is and the type of person he is."
Bar a pre-match presentation to him from Chelsea and hitting the banner bearing his name in the stands with a wayward free-kick, Drogba was largely anonymous in a Galatasaray performance labelled disappointing by manager Roberto Mancini.
It was a far cry from the Drogba seen in his eight-year spell in blue.
The Ivory Coast striker addressed Chelsea's in-house TV channel afterwards alongside Blues captain John Terry.
"It was good, a lot of emotions and a tough game, with good players and good people," Drogba said.
"I was really happy to come back here."
Terry added to Chelsea TV: "He deserved the send off and, after what happened in Munich, I think it's nice for him to come back and for the fans to do what they did and show how much we love him."
Chelsea last season suffered the ignominy of becoming the first Champions League holders to exit at the group stage the following campaign, but responded by winning the Europa League.
Midfielder Frank Lampard praised the impact of Mourinho for re-establishing the Blues among Europe's elite, before warning there was still plenty to do to claim silverware this term.
"He's the best manager in the world," Lampard said.
"You can see the difference this year in the league and the Champions League. The team's strong. He drives the team and he wants to win.
"It's good, but we need to carry on. We haven't won anything yet."