Frei and striking partner Marco Streller said Swiss teams had often lacked self-belief in the past, but this would not be the case when Basel, unbeaten in 17 games in all competitions, defended a 1-0 first-leg lead at the Allianz Arena in their round-of-16 tie.
"We are on a run, we have luck on our side at the moment and we're always good for a goal or two," Frei, the all-time leading scorer for the Swiss national side, told reporters.
"In the last few years, the self-belief in all Swiss sport has improved. We have a 1-0 lead, everything is open and that speaks for the quality of our team."
Basel are attempting to become the first Swiss side to reach the quarter-finals of Europe's top club competition since Grasshoppers in 1979 in the days of the old European Cup.
They have already knocked out Manchester United, qualifying at the expense of the English champions after drawing 3-3 away and beating them 1-0 at home in the group stage.
Frei and Streller, who have both quit international football, are the elder statesmen in a team which includes young players such as Xherdan Shaqiri (20), Granit Xhaka (19), Fabian Frei - unrelated to Alex - (23), Valentin Stocker (22) and goalkeeper Yann Sommer (23).
"Our young players have learnt at a very young age not to be overawed and do not tremble at the thought of facing big teams. That gives us the belief that we can pull off a sensation," said Streller.
"This is the sort of game that comes along once in a hundred years, it's a very rare moment.
"If we can score, that makes it that much more difficult for Bayern. They are a top team, but they are under pressure. It will be very difficult for us, but we have a chance."
Basel scored in every away game in their group and Frei said they would have to do so again if they wanted to go through.
"If we want to go further we must score an away goal," said the 32-year-old who scored 42 goals in 84 appearances for Switzerland and is Basel's leading scorer in both the Swiss League, with 14 goals, and the Champions League, with five.
"We're conscious that we must survive the first wave of pressure and try to deliver a lucky punch," added coach Heiko Vogel, a former Bayern youth team coach. "I won't tell my team just to sit back and defend; we must be courageous."
