Bosnia's Euro 2016 qualifying campaign got off to the worst possible start on home soil after Demetris Christofi's brace cancelled out Vedad Ibisevic's sixth-minute opener in Zenica.
The home side did have a golden opportunity to salvage a point in the 90th minute, when captain Edin Dzeko was fouled inside the area but Pjanic's penalty was saved by Antonis Georgallides.
"I feel sorry for my team and our fans. I took the penalty and didn't score, and I take responsibility for this loss," the 24-year-old said.
"It is true that this is just the beginning of qualification, and that we still have time to fix things, but I'm sorry it ended up this way.
"Right now I can't explain why we played so badly."
Meanwhile, coach Safet Susic lamented Bosnia's complacency after taking an early lead at the Stadion Bilino Polje.
"We didn't deserve to win. We played well for the first 10 minutes and got an early goal. After that, simply nothing worked," said Susic.
"It looked like we were satisfied with an early lead, and having a one-goal advantage against a team like Cyprus is just not enough.
"Then we didn't use some good chances to get another goal. It was a bad day."
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