The reigning Premier League champions headed to Camp Nou with the last-16 tie all square at 1-1, but Messi, who equalised at Stamford Bridge, was on target inside three minutes.
Ousmane Dembele smashed his maiden Barca goal into the top corner after 20 minutes to halt an encouraging Chelsea response and, after Alonso hit the post with a free-kick, Messi nutmegged visiting goalkeeper Thibaut Courtois to net his 100th strike in the competition.
"They have Messi and I think maybe they were the better team even though I think we competed well in both games," Alonso told BT Sport, having also had a strong penalty appeal turned down early in the second half.
"If we concede these easy goals then we make our chances very difficult against this team.
"We knew they have top players and we cannot give them one metre because they can create and score.
"I think we were a little bit unlucky, also, with the first two goals. We hit the post three times [over the two legs], we had so many chances in both games.
"They were the better team but we were unlucky. We competed well and we have to improve. When you lose you have to keep training and get better."
Alonso added: "They were not comfortable in the first half and we just missed the chances.
"It could have been 2-2 at half-time at least but it was 2-0 and in a stadium like this, against a team like this, it is not easy to come back."