A double from Bayer Leverkusen front man Patrick Shick did the damage but McGinn insisted that the Scots should not be too discouraged after failing to convert a number of good opportunities.
The Aston Villa midfielder refuted suggestions that the match was a wake-up call for the national side in their first major finals appearance in 23 years and insisted the squad will pick themselves up for the England match on Friday.
"I don't think it's a reality check," McGinn said. "I think we played pretty well overall. There were things we could improve on, but it's the first game of the tournament and we've got two games to fix what we need to fix.
"It's important we don't sulk too much. We'll deal with it tomorrow, look back at it, and from there we build towards England. We'll need to do it the hard way now."
McGinn admitted that Scotland should have done better with the goals conceded either side of half-time.
"I thought we played pretty well in the first half without threatening them too much, or getting in behind them," he said.
"The first goal is a poor one to lose. A brilliant header but from our point of view we're extremely disappointed with it. You don't want to lose a goal from a set play.
"The second one is a brilliant strike but again we can do better. We can sit here and say we played alright, because we did, and a 2-0 scoreline probably flatters them a wee bit, but they found the quality in the right moments and we need to find that on Friday.
"We can either sulk about it or go to Friday with the same attitude and hopefully get a better result.
"We're here at a major tournament and you need to show your quality. I had a couple of chances - the other boys had chances - and if we want to compete at this level we have to take those chances. Ultimately today we were punished for it."