After a bright start by Arsenal, it was Wigan who went ahead through a penalty from Jordi Gomez just after the hour.
However the Gunners made it 1-1 with seven minutes left through Per Mertesacker, whose foul had given away the spot-kick.
Rosler felt his team could not have given any more. He said: "We faced Arsenal - one of the top teams in the country - and did very well.
"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."
Rosler added: "I got asked 'have you practised penalties?' I said, 'no, I am German, you know we dont need to practice penalties'.
"We want to be back here in May (for the play-offs), when we are back in May, then we will practice penalties."
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