Hazard hit the winning penalty after Chelsea played out a second 1-1 draw with Eintracht Frankfurt, Kepa Arrizabalaga having made shoot-out saves from Martin Hinteregger and Goncalo Paciencia.
Thursday's semi-final may have been the last time Hazard plays at Stamford Bridge, with the Belgium forward close to agreeing a deal to join Real Madrid, according to reports in Spain.
Hazard's contract with Chelsea expires in 2020 but his situation is complicated by the club's two-window transfer ban, which the Blues are appealing to the Court of Arbitration for Sport.
Chelsea would be unable to buy a replacement for Hazard were the sanction to stand and the 28-year-old indicated he remains willing to discuss staying at the club.
"I don't think [about] that. I'm only thinking about trying to win something for this club," Hazard told BT Sport when asked about his future, having helped secure a Baku showdown with Arsenal later this month.
"If it's my last game [the final] I will try to do everything. In my mind I don't know yet, so I will just try to win games for Chelsea."
Chelsea was second best at times in both legs against Eintracht but battled through to a second final of Maurizio Sarri's first season in charge, having been beaten by Manchester City on spot-kicks in the Carabao Cup at Wembley.
"It was hard but when you start a game you think you can reach the final, especially when you draw the first game away," added Hazard. "At home it's a bit easier but today it wasn't.
"I think we played a great game but the semi-final of the Europa League is not easy. At the end we won on penalties and we're just happy. Now we go to Baku and will try to win the trophy.
"In the first half, yes, I think we were the better team, but then as keeps happening we concede a goal after five minutes of the second half. It was 50-50 in the end and penalties is 50-50 as well.
"It's a penalty, you never know, but I always take my responsibility. I scored and the boys are happy, the fans are happy, but it's not finished.
"We have a game to play against Arsenal. We need to go to Baku and play with quality and win the game."