The Blues were pegged back early in the second half, having taken the lead through Olivier Giroud's stooping header.
"We needed a reaction after Saturday's performance but we're disappointed not to get the three points," Azpilicueta explained.
"We talked about showing more character because we were very disappointed with the result and the performance [against Arsenal]. We were far from the level this group has shown a few weeks ago.
"From the first minute, the team competed well. We pushed until the end to get the three points but we couldn't find a way."