Referee Anthony Taylor, who had already booked Costa for dissent in the first half, did not penalise the Spaniard for his allegedly reckless act which according to Blues boss Conte was the correct decision.
"I saw him put pressure on the goalkeeper and then try to stop. In my opinion the referee made the right decision," Conte said.
In the other corner was Hammers manager Bilic, who felt the incident may have gone under the radar, even if it wasn't as bad as many others thought.
"Diego Costa had already been booked," he said.
"It was not reckless or deliberate but it was quite late, the ball had come. He could have been booked again and so a red card.
"I'm not frustrated because of that, I am frustrated because of the late goal."
Following Eden Hazard's penalty early in the second half, James Collins volleyed home for West Ham on 77 minutes, before Costa came up with an 89th-minute winner when firing home from outside the area to give Chelsea a winning start to the new campaign.