The Spain international became a father for a third time on Tuesday evening alongside wife Danielle Semaan.
Leonardo is the latest member of the Fabregas-Semaan family, joining sisters Lia and Capri.
Despite taking in an emotional and tiring Tuesday, Fabregas was back in Premier League action 24 hours later.
He was handed a starting berth by Antonio Conte and lasted 81 minutes before being withdrawn from a crucial 2-1 victory for the Blues.
"It was a massive game and thankfully we won," Fabregas told Chelsea TV afterwards.
"The result (against Crystal Palace) on Saturday put a lot of pressure on us. We know we still have a good advantage, but this is the Premier League and you don't want to be giving points away, especially at home.
"We have character. We can bounce back at anything that comes in front of us. It wasn't the most pretty game from us, but it was a very consistent performance.
"We knew what we were doing, we had a game plan and I think it worked perfectly."
Victory over City has ensured that Chelsea remain seven points clear at the top of the Premier League table.