Both legs ended 2-2 but the Swiss side triumphed on penalties, and Frei said that it is a game that he will remember for life.
"I don't know how I feel right now," Frei told UEFA.com. "You will remember these moments all of your life.
"This is a life of a professional footballer – in three days' time we have another game, so we already have to focus on that. We can't celebrate because the next opponents are just around the corner.
"It's the same as the match last season against Manchester United – the same feelings but the semi-finals for us are even greater. This is the sweetest taste."
Basel go into the semi-final draw with Fenerbahce, Chelsea and Benfica.