Greece coach Fernando Santos said their erratic start - a 1-1 draw with Poland and a 2-1 loss to the Czech Republic in Group A - before their victory against group favourites Russia meant they were neither the worst nor the best team.
The Greeks finished in second place level on four points with Russia behind Group A winners the Czech Republic, who had six points, and co-hosts Poland who came bottom with two.
They only qualified by virtue of a better head-to-head record courtesy of their victory over the Russians.
"We have to clear our heads and keep our feet stuck on the ground so that we can prepare for whatever awaits us in the last eight," said Santos, who took over in 2010 from the revered Otto Rehhagel, the architect of their Euro 2004 triumph.
"Beating Russia does not make us the best team in the world and the bad start we had does not make us the worst team either."
The Greeks conceded only five goals in 10 Euro qualifiers before letting in three in the first two matches at the finals.
"We know who we are and we will give it all we have got. The real character of this team comes to the surface in the tough games," Santos added.
Greece have now lost only twice in their last 24 games since the 2010 World Cup.
They will come up against the winners of Group B where Germany lead the standings with six points and take on Denmark on Sunday needing only a point to top it.
The quarter-final will be held in Gdansk on June 22.
They will have do it without captain Giorgos Karagounis, who equalled the Greek record of 120 caps on Saturday but was booked and is suspended for one match.
"We are not afraid of the big teams in this tournament," said defender Giorgos Tzavellas. "Even if it is Germany, I am not worried. We do really well against the tough teams."
