Vaclik made several impressive saves at Hampden, including a fingertip stop to deny Scotland captain Andy Robertson when the game was still goalless in the first half.
Silhavy heaped praise on the 32-year-old who plays his club football in Spain for Sevilla.
"The performance of Tomas Vaclik was simply excellent,” said Silhavy. “He contributed greatly to our win. He deserves a big thank you from us all.
"The decisive factor was that we survived the early pressure from Scotland. Then we got into the game and scored the opening goal. The second goal from Patrik Schick calmed us down and finished the job."
Bayer Leverkusen striker Schick collected the official man of the match award from UEFA for his contribution as the Czechs made the ideal start to their Euro 2020 campaign and left Scotland at the bottom of Group D.
Schick produced an early contender for goal of the tournament with his stunning second goal scored from just inside the Scotland half of the pitch as he caught home goalkeeper David Marshall out of position.
"I saw him off his line, I checked in the first half when this situation would come,” said Schick. “I was checking where he was standing in those situations.
"It was a fantastic feeling to see the ball fly in, it went in simply perfectly. I ran to celebrate it with our fans. But still, Tomas Vaclik was phenomenal today - he helped us a lot. If Scotland had scored one goal, even when we were 2-0 up, there would have been a very nervous end to the match.
"It is a perfect start for us - a clean sheet with two goals scored. We could have added more, though Tomas Vaclík made several fantastic saves. We showed a lot of teamwork today. It's a very satisfactory start.
"It's always hard to play against us, we have a lot of hard working players. Scotland were a tough opponent. But we were ready for their tactics."