Robben had the chance to put Bayern on their way to European glory in extra time on Saturday when Didier Drogba fouled Franck Ribery inside the box.
But Petr Cech saved what was a poor spot-kick from his former Blues team-mate before going on to help his side win a dramatic penalty shoot-out. Bayern led that shoot-out early on, having also been two minutes away from winning the final in normal time.
Robben, who also fluffed a penalty in their Bundesliga title showdown with Borussia Dortmund, said: "I can't describe how I feel with words, but it's been a terrible night. Two or three times you feel you have the cup in your hands, but at the end you stand with nothing.
"It wasn't a good penalty. I wanted to shoot the ball hard and high in the goal, but the ball didn't go high enough. It was a terrible penalty kick. After that, I still played well but you have to change your mindset at that point."
Robben was consoled by both match-winner Drogba and UEFA president Michel Platini after Bayern's 4-3 shoot-out defeat, following a 1-1 draw at the Allianz Arena. "The fact that Drogba and Platini tried to cheer me up was nice, but it's worth nothing," Robben said. "I wanted that cup but it didn't happen."
Team-mate Thomas Muller, who thought he had scored the winner for Bayern, insisted the better team lost. He told Sky Sports: "That is football, we have seen it in the past. It is not always the better team that have the cup in the end.
"When you score in front of your own fans so close to the end and then you watch Chelsea holding the cup, you can see that a lot has happened. It is hard to find words for that, but football is not everything."