Cole hailed his team's self-belief.
As well as John Terry's red card, the visitors were also disrupted by the loss of Gary Cahill through injury.
Cole said on Sky Sports 2: "When you lose quality centre-halves like that against a great team, it's going to be hard.
"But we all believed. Not many people did but we did as a group of players, and that's what happened.
"It's hard, but we defended well as a team, got a bit lucky, and Petr (Cech) made some good saves.
"Three to four months ago no one thought we had a chance. We lost a few games in the Premier League but we always believed and knew we were good enough to be here. You can't beat that desire and fight that you saw out there."