The hosts appeared well placed to pip Atletico by virtue of a superior head-to-head record when Alexis Sanchez's blistering strike gave them a 34th-minute lead.
But Atletico, shorn of key men Diego Costa and Arda Turan through first-half injuries, flew out of the blocks after the restart and levelled decisively through Diego Godin's header.
Barca could not find the goal they needed to win the league and, after rising to acclaim the new champions, the home supporters whistled and jeered their own players at the end of a trophyless season that looks set to cost coach Gerardo Martino his job.
"Atletico are deserved champions," Iniesta told reporters.
"But it's been a disappointing season for us."
Barcelona sporting director Andoni Zubizarreta echoed Iniesta's sentiments when it came to the Madrid outfit but refused to label his own team's efforts as a failure
Zubizaretta told reporters: "Congratulations to Atleti."
"It wasn't that Barca failed; we were up against really tough opponents."
