Costa scored the decisive goal in Chelsea's 1-0 win at Arsenal on Sunday, not long after being tripped by Per Mertesacker, who was sent off after 18 minutes.
The striker now has six goals in six games under Hiddink, who hopes he will shake off a knee problem which forced his early withdrawal at the Emirates Stadium.
Arsenal boss Arsene Wenger again bemoaned the influence of Costa, who antagonised the Gunners last September when Gabriel saw red.
But Hiddink was grateful Costa focused on football and delivered.
"He is important, because he's focusing on his first job," the Dutchman said on Chelsea TV.
"If you have to play him as a central defender, then you are a bit in trouble.
"He likes to fight - in a good way - and he's got his credentials."
Costa was one of the players who drew the Chelsea supporters' anger after Jose Mourinho's departure last month.
Hiddink reckons Chelsea's players have won back the faith of the supporters after overseeing a seven-match unbeaten run.
Chelsea are undefeated in eight since Mourinho's departure, the first game of that sequence a defeat of Sunderland prior to Hiddink's appointment which was saw the fans turn on the players who seven months earlier had delivered the title.
Like Costa, Cesc Fabregas was one of the primary targets for the supporters' frustration then, but both excelled at the Emirates Stadium, with the playmaker appearing to thrive on the taunts against his former club.
Hiddink added: "When I came in I said 'hey guys, you have to win back the support and the confidence of the Chelsea supporters and then we're looking to the future'.
"With the commitment of the players, on top of the way they played, they are pleasing the fans again and the fans are closing them in their arms."
Hiddink expressed admiration for Arsenal's style, but praised his men for not allowing the Gunners to play before calling on Chelsea to be more consistent.
"I like to see them play, but our credit was not to have them play," he added.
"We were having an unbeaten run of games, but there were some draws and also some games where we had to come back.
"The team reacted very well, but that's not enough
I want a team that's not having this negative score.
"We want to be proactive
We must be more consistent in getting the wins."
Midfielder Nemanja Matic added on chelseafc.com: "This was the real Chelsea, and I believe that we can show again in the future that we deserve to play for this club.
"Of course we are not in a good position in the table but there are a lot of games until the end of the season and we have chance to show our quality.
"I am sure that this win gives us more confidence for the future.
"We deserved to win, we showed our qualities and that we can play good football, and I am sure that until the end of the season we will show that we are still a big team and that we can win something."
Chelsea are 13th in the table, 14 points adrift of fourth-placed Tottenham and the Champions League qualification places with 15 games to play.
Their next fixture is the FA Cup fourth-round tie at Milton Keynes on Sunday.
Source : PA