The Spain midfielder says his side have squandered opportunities throughout the season as Sunday's 1-1 draw at Camp Nou handed Diego Simeone's team the championship
Diego Simeone's side sealed a 1-1 draw in a thrilling showdown on Sunday, with Diego Godin's header ensuring they finish three points clear of last season's champions in the table.
Barca led for the first 50 minutes thanks to Alexis Sanchez's stunning strike but were unable to find a winner and Iniesta believes they paid the price for failing to take their chances.
"It was a sad end. We had it until the end. We have to congratulate Atletico. They are deserving champions," he told Canal Plus.
"We have failed, as a group. We had chances until the last match, like them. Atletico have been better, and are champions. There is nothing to say."
Speculation has persisted this week over the future of coach Gerardo Martino, with Luis Enrique - widely reported as a prime candidate to take over at Camp Nou - stepping down from his post at Celta Vigo.
Iniesta, however, insists Barca cannot look for excuses after surrendering their domestic crown.
"It hasn't affected us, thinking about other things these last weeks. It's talk for talking. It hurts losing this league, nothing more," he added.