A 2-0 win on the night in Germany was not enough to stop the Gunners exiting the competition with a 3-3 aggregate scoreline.
"We really thought we could do it," Jenkinson told Sky Sports News.
"I'm proud of the boys and the effort we put in. We've come here and won 2-0 and were close to doing it."
The Gunners return to Premier League action on Saturday when they travel to Swansea City.