The Argentine forward scored twice at Camp Nou, as the Blaugrana won 4-0 to progress to the quarter-finals 4-2 on aggregate.
"We needed a night like that not only to get through, but to erase that result in Milan," he told UEFA.com. "Everything came off perfectly.
"I think our fans knew just how we were going to respond and we, the players, knew the crucial thing was to re-establish our way of playing. We did it. From the fifth minute when we got that important first goal, we just went after Milan in every way.
"We wanted that opening goal as soon as possible to cut the deficit and to make the fans feel it was going to happen. It was lightning quick movement of the ball to get it to me and, happily, I put it away. That's why we celebrated with such joy."
Messi's double made him the second-highest goalscorer of all time in the Champions League.