Scotland and Denmark meet on Tuesday in their final group tie with the winner quallifying directly for next summer's finals.
"The aim of tonight was to make sure we had a cup final on Tuesday and we have it, in strange circumstances," McGinn told BBC Scotland.
"So this opportunity, we can't let it go."
Scotland were beaten on Saturday by Greece 3-2.
McGinn said it "shouldn't have taken us to 3-0 down" to produce the performance they mustered in the final 25 minutes.
"The manager said the last 20, 30 minutes was everything that we've shown as a group over the past five, six years which got us some success, which got us to tournaments - direct, full of energy, brave, creating chances," the Aston Villa captain explained.
"What an effort we put in towards the end. It feels a bit weird, the feeling of relief that Denmark didn't win gives us an opportunity for a cup final on Tuesday night.
"We've showed signs of a better performance, but there's still a lot to learn, we know that.
"It did baffle me a wee bit, there was only four minutes added, but you can't really expect this when you're away from home. You don't really get the rub of the green. For us, we can see the three poor goals, but what we showed tonight is character and abundance to come back to 3-2."