The Blues opened the scoring in the game through Eden Hazard's penalty in the 47th minute but were on the verge of dropping points after James Collins equalised against the run of play in the 77th minute.
However, Diego Costa scored a winner in the 89th minute to claim all three points for the West Londoners much to the delight of Azpilicueta.
"We deserved to win, we didn't deserve to concede the goal and to get the spirit and confidence back from the equaliser was a tough point but the team reacted very well," he told the club's official website.
"Our target was to start the second half strongly because in the first half we played well but we couldn't get the goal everyone was looking for.
"Our main target was to control the game, to press and recover the ball as quickly as possible. I felt we played well with the ball; we had a lot of possession and created a lot of chances. We recovered the ball really high and that was a key point which we did really well."