The North London club almost overturned their 3-1 deficit from the first leg by winning 2-0 at the Allianz Arena but last season's runners-up Bayern edged them on away goals after the two-legged tie ended 3-3 on aggregate.
Defender Jenkinson was pleased with the performance of the Gunners and suggested that plenty of pride was flowing through the dressing room following the match.
"We got that early goal that we needed. We knew that it was important that we kept it tight at the back and we did that. We got the second goal and we really thought we could do it, but I am proud of the boys and the effort everyone put in," he told Sky Sports.
"We were so close to doing it when, to be honest, everyone had written us before the game. We have come here, we have won 2-0 and come so close to doing it. I am very proud of the boys."