Bayern Munich reached the Champions League quarter-finals for the seventh successive year with a thumping victory over Besiktas.
While Courtois did not take full blame for the result, he said he should have done better to stop Messi's opening goal, which came just 128 seconds into the game.
Conte offered no guarantees, adding: "I think that we are talking about two good goalkeepers, Thibaut and Willy".
Messi struck after 126 seconds, netting through the legs of the hapless Thibaut Courtois, and struck his 100th Champions League goal, again through the Chelsea goalkeeper's legs.
Courtois admitted being at fault, telling BT Sport: "I didn't expect Messi to shoot from there and I was too late to close my legs so that was a mistake on my part".
"We ask that our supporters contact us with accounts of their experiences of this evening's arrival at the stadium so that we can take this up properly with the authorities". "Tomorrow I will pick the best player because we want to go to the next round", Conte added.
'We have to be honest about that.
"I played against Messi a lot of times, I have already conceded goals like that before". It's my weakest point between my legs.
"It is annoying, I can not hide inside - I have to be a man and come out".
By the end, the Argentine's scintillating breakaway in the build-up to Dembele's first Barcelona goal had become little more than a footnote to what was another Messi masterclass. "In the last game we played at home we suffered against them", Conte told reporters on Friday.
Busquets, 29, has played in all of Barca's Champions League games this season and only missed three of their league games. We have to improve and, when you lose, you have to train and get better. "We had plenty of chances". There are no easy teams at this stage and Barcelona are the favourites in the tie but no one should write Roma off.