The Blues carried a 1-0 deficit into the second leg of the semi-final at Stamford Bridge but goals from N'Golo Kante and Hazard were their reward for a stirring first-half display.
Fernando Llorente's header restored aggregate parity but Chelsea converted four out of four penalties as Eric Dier and Lucas Moura erred for the visitors on Thursday.
Sarri questioned his players' mentality and appetite for the battle after going down 2-0 to Arsenal in the Premier League at the weekend – words that certainly seemed to bring a collective response – while he called Hazard an "individual" as opposed to a "leader" at his pre-match news conference on Wednesday.
Asked whether he was affected by Sarri's observations, Hazard broke into a broad grin and told Sky Sports: "No, to be fair I don't care.
"It doesn't matter what the manager says. I just want to do my best for this team and I'm happy."
46 - Chelsea have lost none of the 46 games (all comps) in which Eden Hazard has scored in at Stamford Bridge (P46 W39 D7 L0). Charm. pic.twitter.com/gjjAAvIVED— OptaJoe (@OptaJoe) January 24, 2019
Chelsea lost 2-0 to City when they met at Wembley in the Community Shield at the start of the season but won by the same scoreline to hand Pep Guardiola's reigning champions their first Premier League loss of the campaign last month.
"It's always hard. They are the best team with Liverpool in the Premier League at the moment," Hazard added.
"They've won a lot of trophies but we also have a lot of great players, so it's a 50-50 game."
Chelsea centre-back David Luiz dispatched the decisive penalty against Tottenham.
"It's a great feeling," he told Sky Sports. "I think the team did great, especially in the first half we could score three or four goals.
"In the second half we controlled the game. They had one opportunity and they scored. You saw the reaction of the team and the spirit of the team."