Costa cancelled out Gary Cahill's own goal opener and set up Willian for Chelsea's second goal as Conte's men came from behind on Saturday to record an eighth successive victory.
The combative Spain international even acted as a peacemaker during an injury-time melee in which City pair Sergio Aguero and Fernandinho were sent off, belying his reputation as a firebrand.
"Diego is showing he is using his passion in the right way and I'm very happy about that," said Conte, whose team put four points between themselves and Pep Guardiola's City.
Costa has frequently fallen foul of England's football authorities since joining Chelsea from Atletico Madrid in 2014.
His form slumped last season as Chelsea's title defence disintegrated, but he is the Premier League's joint-top scorer with 11 goals and believes he has benefited from keeping his head.
"I want to be on the pitch and I want to be cool," Costa said.
City deservedly went ahead on the stroke of half-time when Cahill's near-post lunge saw him turn Jesus Navas's cross into his own net.
Costa equalised on the hour, chesting down a pass from Cesc Fabregas and slamming a low shot past Claudio Bravo, before freeing substitute Willian to run through and score Chelsea's second.
Eden Hazard sealed victory in the final minute of normal time after cantering onto a pass from Marcos Alonso.
An injury to Nemanja Matic saw Fabregas make his first appearance since Chelsea's 3-0 loss at Arsenal on September 24.
Chelsea missed Matic's positional discipline as City pulled them all over the pitch, but Conte saluted Fabregas's contribution.
"He played a good game," said the Italian. "I know I can always count on him."
City took the lead after a sustained spell of pressure and might have been awarded as many as three penalties.
Kevin De Bruyne squandered a glorious chance to make it 2-0 in the 56th minute, hitting the bar from barely five yards out, and four minutes later Chelsea levelled.
Guardiola defends Aguero
The main talking points in an absorbing game occurred in added time, when Argentine striker Aguero was red-carded for launching himself into an appalling knee-high challenge on David Luiz.
Guardiola, implausibly, said the foul was not intentional.
He will be without Aguero for four games, while Brazilian midfielder Fernandinho faces a three-game ban after grabbing Fabregas by the throat and shoving him over an advertising hoarding.
Guardiola said Fernandinho had been trying to "defend his team-mate" and drew a confusing parallel with Nolito's dismissal for headbutting Bournemouth's Adam Smith in September.
"It's a pity it finished like this, but it was the way the referee decided," said Guardiola, who tasted defeat at the Etihad Stadium for the first time.
"I don't like that. I apologise for what happened."
City's fans were incensed by referee Anthony Taylor, who elected not to punish Luiz for impeding Aguero when he was clean through on goal and also turned down three penalty appeals by the hosts.
But Guardiola said: "We didn't win because we missed a lot of chances. Not for the referee."
He added: "I am so proud of my team and how we played.
"I came here to try to play the way we played today. We created more chances than against Burnley and Crystal Palace, when we won.
"In the boxes, we are not strong enough. But I have no regrets."
Conte congratulated his players for keeping their heads.
"I'm happy because I always tell my players to be focused on the game on the pitch and always to keep calm," he said.
"When you play these big games it's not easy to keep calm in every single moment, but I'm pleased because all my players tried to keep calm."