The uncompromising forward was a menace throughout Tuesday night's Capital One Cup semi-final second leg, but could miss Saturday's Premier League title bout with Manchester City through suspension following apparent stamps on Emre Can and Martin Skrtel.
"When you play with this guy, you have to give everything," Hazard said of Costa.
"You can see in every action and for every ball, he gives his life.
"Even if he didn't score against Liverpool, he gave his life
For us, for the players, when you play with him, it's very good.
"In training he is a little different but in the game he is very good
He never stops
He had a very good mentality against Liverpool.
"We need him because he can score every minute, every second
He did very well."
Jose Mourinho defended Costa's footwork as "absolutely accidental" after the match and reserved particular praise for Branislav Ivanovic, whose white boot was blood-stained and should be showcased for Chelsea's young players as an example, the Blues boss said.
Ivanovic's extra-time header earned a 2-1 aggregate win and a place in the March 1 final at Wembley.
Sheffield United or Tottenham, who play on Wednesday night, will be the opponents, with Hazard hoping Chelsea can make amends for the 5-3 New Year's Day loss to Spurs.
The Belgian said: "Yes, to get revenge - it'd be good
We have to.
"But it doesn't matter if it's Tottenham or the other one, we have to win the final."
Serbia defender Ivanovic believes Chelsea showed their true force against the Reds and warned Manchester City there is more to come.
"We showed the real power of our team and in the end deserved to be in the final," Ivanovic told Chelsea TV.
"We know we can improve and we'll improve in the next couple of games."
Chelsea bounced back from the shock FA Cup elimination to League One Bradford as Mourinho made nine changes to his starting line-up.
Worryingly for Gary Cahill, he was not in Chelsea's team, dropping to the substitutes' bench after a series of below-par displays, including against Bradford.
Kurt Zouma stepped up and performed well against Raheem Sterling, who scored Liverpool's first-leg goal at Anfield as Cahill backed off.
"Nobody enjoys to play against Sterling, because he is so, so quick," Mourinho told Chelsea TV.
"Zouma was very, very focused, very concentrated on the job he had to do
Zouma did fantastic."
With City visiting Stamford Bridge on Saturday, Premier League leaders Chelsea could have done without the extra half hour of play.
The Blues hold a five-point lead which could be increased to eight, or fall to two, on Saturday evening.
Mourinho played down concerns of tiredness and thanked the supporters after the humiliating loss to Bradford.
He said: "The players are tired but when you win fatigue doesn't mean the same as when you lose.
"We have a big game on Saturday, but I think the feeling of winning, being at Wembley, compensates a little bit the fans after the gift we gave them last weekend.
"For our fans, after Saturday, the best thing to happen is now we go to Wembley."
The tie might have been settled sooner had the goalkeepers not made crucial interventions.
Thibaut Courtois saved well from Alberto Moreno and Philippe Coutinho, while Simon Mignolet twice denied Costa.
Liverpool's trophy hopes now rely on the FA Cup, with Bolton waiting in a fourth-round replay.
First the Reds play West Ham in the Premier League, where a top-four spot and Champions League qualification is the target.
Mignolet told liverpoolfc.com: "The most important thing is that we have to go again on Saturday in the Premier League and then the FA Cup after that
We have to pick ourselves up and go again.
"Everybody is disappointed when you go out of a competition that you want to win, and especially with the effort and performances that we put in over the two legs.
"We definitely deserved more than we got
But we have to accept it and move on, and make sure that we go again in the other competitions that we're still playing in."
Source : PA