The Azzurri drew 1-1 with Spain at Wembley but were dominated in terms of possession, having about 35% of the ball and shots, with seven to the their opponents' 16, before triumphing 4-2 on penalties.
"There are games where you have to suffer," Mancini told a news conference. "They can't all be as smooth as our progress so far. We knew it would be a tough game.
"That is why the players and everyone who worked with us over last three years deserve great credit, because it wasn't easy.
"Almost no one believed we could do it, and yet we are into the final. We are pleased for Italians everywhere who have been following us for the last few weeks."
Italy will face Denmark or England in the final on Sunday as they bid to win their first European Championship title since 1968. It will be their first final appearance since a 4-0 defeat to Spain at Euro 2012.
However, the Azzurri were often on the back foot against a talented young Spanish team that starved them of the ball, and Mancini was eager to underline the scale of his side's achievement.
"We are delighted to have provided this wonderful evening's entertainment to the Italian people," he said. "We still have one game to go.
"We knew it was going to be a very tough match, because in terms of ball possession Spain are the best around. They caused us problems, we had to dig in when we needed to, we had to score and create opportunities where we could.
"There were some issues because we did not have much possession. But we wanted to reach the final and kept trying until the end. Penalties are a lottery, but I want to take my hat off to Spain; they are a wonderful team."
Winger Federico Chiesa was awarded the UEFA 'star of the match' award after scoring a superb curling opener, and the 23-year-old was asked if it was the highlight of his career.
"Without a doubt," he said. "Playing for my country in matches like these, representing 60 million Italians, was an unbelievable dream that I never imagined."
Asked for his thoughts on Italy's potential final opponents, he said: "Denmark play with a lot of intensity and throughout this tournament they have proven what a top side they are; they have some quality players.
"As for England, they have shown that their individual players can be match winners. They have some top players in the final third and all across the pitch."