The openers put on a 98-run partnership in the win over Bangladesh on Tuesday, but were unable to produce their best when it mattered most as the pre-tournament favourites failed to reach the semi-finals.
"Myself and Davey have both been disappointing in this tournament as a partnership," Finch told reporters.
"What makes it even more disappointing is that it took until the final game to have a big partnership. To be out of the tournament already and then provide a start for the team is very disappointing.
"It's frustrating because we came here with high expectations. I don't think we played particularly well in the first three games and in such a short format you have to rely on your openers. So we take a lot of responsibility."
Finch is currently ranked as the best batsman in Twenty20 international cricket by the ICC.