Bayern put on a spirited display at Real Madrid on Tuesday but were ultimately denied a place in the final, drawing 2-2 to lose the tie 4-3 on aggregate.
Ribery played the full 90 minutes at the Santiago Bernabeu, where he was left disappointed in his bid for a second European crown.
The Bavarians wrapped up their sixth straight Bundesliga title at a canter and could complete a domestic double when they take on Eintracht Frankfurt – the team managed by incoming Bayern boss Niko Kovac – in the DFB-Pokal final on May 19.
Ribery says Tuesday's setback will only make the players more determined to win the cup for a 19th time.
"We came to Bernabeu to make it to the final," he tweeted. "We gave everything we had and still the journey ends here. Thank you for all your support from near and far.
"There is another important final left this season and we will play with all our hearts to get that title for you!"
Bayern last claimed the DFB-Pokal in the 2015-16 and have won three of the previous five crowns.
Meanwhile Eintracht is looking to claim their first trophy in the competition since the 1987-88 campaign. However, the club did make the final last season, ultimately falling to Borussia Dortmund by a 2-1 scoreline.
This year's finalist met last week in Bundesliga action, with Bayern coming away 4-1 victors.