The Blues have won the FA Cup four times in the past 10 seasons, but not since Hazard moved to Stamford Bridge in June 2012.
The in-form Belgian, who has lifted the Premier League, Europa League and League Cup with Chelsea, is now hoping to end that wait.
Speaking ahead of Monday night’s quarter-final clash with Jose Mourinho’s Manchester United, Hazard told Chelsea TV: “The FA Cup for all the fans does mean something special. They won a lot in the past 10 years.
“Since I signed for Chelsea I never won this cup.
“The FA Cup is something we want to win, but when you play for Chelsea it is not something you think about a lot because you want to win every game.
“My target is to give everything and only if we win the trophies will I be relaxed.”
Hazard sparkled in former Chelsea boss Mourinho’s title-winning side of 2014-15, but, hampered by a niggling hip problem, performed poorly last term as the Blues’ title defence unravelled.
That was until Chelsea drew 2-2 with Tottenham, with Hazard’s equaliser confirming Leicester as champions.
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Chelsea’s bid to reclaim the Premier League title this season – they are 10 points clear at the summit – is on hold as United visit, but Hazard wants Antonio Conte’s rejuvenated side to “give everything” in search of silverware on two fronts.
“It is a key moment in the season and if we want to be champions we have to give everything,” Hazard added.
“All the games from now until the end of the season will be difficult because everyone wants to beat the team that is at the top of the league, so we have to be ready for that.
“All the opponents will give everything and we have to do our best.”