Alvaro Morata cancelled out Ivan Rakitic’s early opener, but goals from Luis Suarez and Neymar sealed a 3-1 victory for Luis Enrique’s men in Berlin.
Pogba was disappointed with the defeat, but he told Sky Sports everyone gave it their all.
“We tried our best but Barcelona were better than us,” he said.
“They had something that we didn’t have tonight and congratulations to them.”
Morata’s goal 10 minutes into the second half looked to have injected life into the final after Barcelona dominated for large spells.
But it wasn’t to be for Massimiliano Allegri’s men, who were also going for a cup treble, and Pogba admitted they felt they could turn the game around.
“When we scored, we thought that we could win the game,” he added.
“And after they scored the second one, our heads went down, but it was never finished.
“We tried and tried again but they scored the last one and that was it.”