Willian, denied twice by the woodwork in the first half, gave the Blues the lead after 62 minutes with a pinpoint drive from the edge of the penalty area.
Chelsea looked set to hold firm against Barca and take a one-goal lead into the return leg on March 14 at Camp Nou, only to shoot themselves in the foot.
Andreas Christensen gave up possession with a sloppy pass deep in his own half, and Andres Iniesta fed Messi to score his first goal in nine appearances against Chelsea.
Conte was left frustrated by Chelsea's failure to close out the victory. Messi's strike was with 15 minutes to go and was his only shot of the game, yet the Italian was delighted with the way his team delivered a defensive masterclass - barring one costly slip.
"I think we were very close, very close to a perfect game," Conte told BT Sport. "I think only one mistake and when you make a mistake against Barcelona, players like Messi, [Luis] Suarez, Iniesta, you have to pay.
"It's a pity, I'm very proud of my players for the game, they followed our plan. It's a pity. It's a pity because also I think tonight we were a bit unlucky.
"I think when we have to have a plan for this game, respect the characteristics of the opponent and understand the best way to face them. Our plan was really really good and the effort was great, it's a pity because we are talking about 1-1 and maybe we deserved more tonight."
Barcelona's away goal gives them the edge ahead of next month's second leg but Conte believes Chelsea can cause an upset when their rivalry renews with the Catalans at Camp Nou.
"This team is very strong, a fantastic team, but I think the result means the tie is open," Conte said.
"It will be difficult, it's not simple to play away against them, but our performance will give us confidence to try and do something."