Walcott was introduced on 66 minutes after his side had gone 2-0 down to Zlatan Ibrahimovic's second goal.
He scored within three minutes before injured skipper Fabregas stuck away a penalty.
The winger said: "It was good to show what I can do. It's been frustrating not starting more but we have a lot of experience now.
"In the past, like against Manchester United last season when we lost 3-1, we dropped our heads. But now, the spirits stay high.
"We did well to hit back with two goals. I thought 2-2 was a fair result after all the possession we had late on."