The England captain, married with two young children, was named on Friday as the player behind a gagging injunction to stop the publication of the stories following a High Court hearing in London.
After the order was lifted newspapers published pages of details about his alleged relationship with a "famous" woman.
Terry, booed every time he touched the ball by the home fans during Saturday's Premier League match, forgot his troubles to head the winner eight minutes from time in a 2-1 victory.
Asked after the match if he had any thoughts about leaving Terry out of the side because of the publicity, Ancelotti told reporters: "No, never, never."
"It was a good moment for Terry on the pitch. But he also played well (in the 3-0 win) against Birmingham in midweek.
"I did not speak with him. It wasn't necessary because he did well in training yesterday. It was not in my mind he wouldn't play.
"I do not want to talk about his personal life. He is doing his job very well, every day, every training session. For this I am not worried.
"All the club, the players, management and staff are supporting him and his family at this time."
"Its a difficult place to come on a cold night, but we did pretty well and deserved to win the game - we created quite a few chances," said Lampard.
"Burnley are a tough team to play against but at this stage of the season with the big game coming up tomorrow it was important we won and we did it."
He told ESPN: "We got back into the game after showing too much respect to Chelsea in the first half, but I thought in the second half we were terrific.
"The disappointing thing is you give them half a chance and they score. I was disappointed not to get a point."
The defeat stretched Burnley's run without a win to 12 league matches and after a bright start to their first season in the top flight since 1976, they remain in the relegation zone.
