Willian's goal had the Premier League title-holders on track to take a 1-0 lead into next month's second leg of the last-16 tie between the old rivals.
But after Azpilicueta's defensive colleague Andreas Christensen gave the ball away deep in his own half, Andres Iniesta set up Lionel Messi for a crucial away goal.
With Gary Cahill on the bench, Azpilicueta wore the captain's armband against Barca and the Spain international wants a flawless display from his side for the second leg on March 14.
"The away goal obviously gives them a small advantage because with a 0-0 they go through, but it is what it is," Azpilicueta told Chelsea's website.
"There is still a lot to play for and it will be an interesting game, we know it's going to be tough but our performance showed what we are capable of doing and we will try our best to qualify.
"We have to play a perfect game, we know we are playing against very good players in a tough stadium but we are confident and we will try to qualify."
"Against Barcelona it was also a great performance from the team, nearly perfect. Now we have a tough game on Sunday and we have to turn our heads to the next target, the Premier League, and try to get as high as possible.
"The next game is always the most important because it's where you can get the three points. At the moment we are getting closer and closer to the end of the season, so we have to get the points."