Wigan held out for a further 30 minutes, but they failed to overcome the Gunners on penalties, putting an end to their cup run.
"We were only eight minutes away from winning, but we had to make some substitutions early and unfortunately we could not hold out," Rosler is quoted as saying by Sky Sports News.
"We forced Arsenal to play some long balls in the second half, which is unusual for the way they play, and we still looked to attack them when we could. I am absolutely proud of how they performed and what they put in.
"When you can see the finish line, but just not get over and then bounce back in extra-time, that shows mental strength and character. When you are then going into penalties, there is always a bit of a lottery."
Wigan turn their attention back to league action on Friday when they face Reading.