The Gunners initially fell behind in the tie as Jordi Gomez fired home a penalty and the tie looked bleak for Arsenal before Per Mertesacker's late equaliser forced extra time.
With no winner found during 30 minutes of extra time the game went to penalities where misses from Gary Caldwell and Jack Collison proved costly for the Latics.
Speaking about the game, Rosler said: "We were only eight minutes away from winning, but we had to make some substitutions early and unfortunately we could not hold out.
"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.
"I guess a lot of people will say 'now they will never recover mentally from this', but we will just use this now - last season at Brentford, we lost on automatic promotion in the 95th minute (of the final day) and the team recovered to go to the Wembley final.
"It is exactly that mental strength, that willpower and desire to get through the play-offs, that is definitely our target, that is what we are breathing, eating and sleeping, and I think that will happen."
The Latics return to league action next and the hunt for promotion back to the top flight as Reading are the visitors to the DW Stadium on Friday in a real play-off six-pointer.