Unai Emery's side were 2-0 down inside 30 minutes at Villa Park but levelled through Ollie Watkins and Moussa Diaby before the visitors scored twice more in the second half to secure the advantage ahead of next Thursday’s return leg in Athens.
"We have a mountain to climb but we know we're capable," McGinn told reporters. "They played very well tactically.
"Everything that could have gone wrong tonight did.
"The manager was still positive in (the dressing room). There's still a long way to go."
Villa manager Emery felt his side lost possession too easily but said Olympiacos deserved the victory.
"I'm frustrated, really frustrated, but I have to accept it," said Emery, who won four Europa League titles with Sevilla and Villarreal.
"We built here at home one style, one idea that’s so, so strong. Of course, sometimes we can lose but in a key moment to get something in Europe, we lost a good opportunity.
"Hopefully next week we can feel better, play better and try to play with personality without losing the ball like today," he added.
Villa will look to tighten their grip on fourth place in the Premier League when they travel to Brighton & Hove Albion on Sunday. They are seven points ahead of Tottenham Hotspur, who have a game in hand.
"We need calm. We are playing at Brighton on Sunday and it's our focus 100 per cent. We have three games left in the Premier League," added Emery.
Should Villa hold onto fourth spot they will qualify for next season’s Champions League for the first time.
They won the European Cup in 1982 and competed in the competition again the following year but have not participated in the continent’s top club competition since.
