The Belgium international came off the bench and scored the winning penalty after the Championship visitors took the tie all the way, doing so with an equaliser in the 94th minute of normal time.
Chelsea, reduced to nine men for the closing stages of extra time after Pedro and Alvaro Morata were dismissed, struggled in front of their own supporters and Hazard admits that it was always going to be a case of simply getting the job done by any means.
"Today was a replay and with a replay you have to win the game," he told BBC Sport at full time. "With extra time and penalties it's not easy. We got two red cards which is the bad thing for today but at least we go through and we are ready for Brighton at the weekend.
"We can focus on the league and also be ready for the next round against Newcastle. We had chances [to kill off the game] but the FA Cup is always not easy. The first game away was a bad game and also today it was not our best game of the season, but we gave everything at the end of the game when it was nine versus 11.
"We showed great character after the draw at the end of time so we are happy, but also sad that we lost two players. But we are ready for the next step."
Chelsea have now been held in five successive matches since the turn of the year, with three of those ending goalless.