The Czechs were never really in the game at the National Stadium in Warsaw and a Cristiano Ronaldo header in the 79th minute was all it took to put the Portuguese through to a semi-final against the winners of Saturday's match between France and Spain.
But Bilek said the disappointment of defeat was tempered by the achievement of taking a side with a blend of youth and experience, and without midfield playmaker Tomas Rosicky, as far in the tournament.
"Immediately after the match we were disappointed because we lost," said Bilek. "We were really looking forward to playing in the semis and the final and we wanted to make it happen. We played well in the tournament. We were playing against a more dangerous opponent and they had more chances. I just want to thank my lads for their performances."
The Czechs have met Portugal six times in major competitions, including a quarter-final meeting in EURO 1996, when a Karel Poborsky winner took them through to the last four and ultimately a final that they would lose to Germany.
However, with a current team largely devoid of big names, apart from Chelsea goalkeeper and team captain Petr Cech, as well as Arsenal's Rosicky and promising striker Vaclav Pilar, few gave Bilek's side much of a chance to progress.
The 47-year-old coach described his time in charge as a rollercoaster and admitted that their achievement had surpassed his expectations, showing great character to fight back from an opening 4-1 group stage reverse against much-fancied Russia.
"If someone had told me that we were going to go to the quarter finals at the EUROs... I don't think I would have believed you. But I know my team. They're strong and I know they're good," said Bilek, who faced calls for him to stand down last October.
"Over the last two years we've created a team that's got great qualities and relations and I'm really happy that we made it this far, especially after the first match. We showed our character and strength. We always miss [Rosicky] when he's not playing because he's an exceptional player, but the whole team has great character. We managed to go on and I have to say that I'm proud of my team."
As well as accepting that the better side won on the night, Bilek said that fatigue played a part. "In the first half we managed, but in the second we were getting tired, so it was easier for them to get behind the defence and they just scored and that was it," he said.
"We had enough days' rest. We weren't that tired after the match against Poland (last Saturday). But it's difficult to play without the ball. You lose strength physically and mentally. You have to track back. That's why the opposition took advantage. They knew how to hang on to the ball and that's what counted."