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.
"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."
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