Before the tournament started, Italy coach Antonio Conte played down expectations for his side, who went out of the 2014 World Cup in Brazil at the group stages.
Italy are short of young talent with just seven players aged 25 or under in the 23-strong squad. But they opened their account in Group E with a 2-0 defeat of Belgium on Monday.
Tavecchio told reporters on Wednesday: "Our wish is to reach the final in Paris against France. I've said it before, Italy play to win and not to arrive second. We have big dreams and not just now because we have beaten Belgium."
Goals from Emmanuele Giaccherini and Graziano Pelle gave Italy a perfect start in Lyon.
"We played and won like a big team," Tavecchio said. "The satisfaction of winning our first game and the awareness of the team's strength remains as we look ahead to our next game."
Italy won the European Championship on home soil in 1968 and were runners-up in 2000 and 2012. Tavecchio praised Conte's role at the 2016 tournament.
"Conte is doing something extraordinary," Tavecchio said. "We have no stars, this is a team that will not suffer internally from jealousy.
"We have players who train hard day after day. The journey is long but I think. After the negative judgements that were made prior to the team's start to the competition, many people must regain belief."
Italy's next group game is against Sweden on Friday before their final round-robin clash with Ireland on June 22.
(Writing by Adriana Garcia, editing by Adrian Warner.)
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