Hazard and Costa were on target to sink Southampton 2-0 at St Mary's and make it four straight league wins for the Blues.
Crucially, Chelsea have not conceded a goal during that run, ever since manager Antonio Conte reverted to his favoured back-three system.
But centre-half Cahill says the defending from the front has been pivotal, with both Hazard and Costa tracking back and tackling impressively against Saints.
"Southampton's record at home is very good and we had to dig in at times," he told Chelsea TV.
"We have worked hard in training
Every player has worked hard on the shape the manager has brought in and we are getting the rewards.
"But it is all right having the shape, it is also to do with having the players with the mentality to go out and do the business when we need to.
"We have the flair players to go out and create and score goals, but we also need them to dig in and defend well, and we did that."
Just one point separates Chelsea in fourth and leaders Manchester City with the title race already shaping up to be the most exciting in years.
"That is where we want to be, up and among it," added Cahill.
"I can't see any team running away with it this year and we want to be in there, we want to be involved."
Southampton have also been in good form but slipped to their first home defeat of the season.
They face Inter Milan in the Europa League at a sold-out St Mary's on Thursday and midfielder Oriol Romeu knows they need to bounce back quickly.
"We don't want to lose two in a row," Romeu told the club website.
"The season is very long and we need to come back
We have been very good recently, and the feelings are bad because we have lost.
"We are being honest with ourselves, but we need to look forward and we need to be proud as well."
Source : PA