Is the prayer valid if one rak'ah of it occurs one minute before sunrise, knowing that the muezzin made the call to prayer before it?

If you say the opening takbir (Allahu Akbar) before the time of Fajr ends, then your prayer is valid and counts as ada’an (performed on time). However, if it is after sunrise, then its validity is as qada’an (a make-up prayer). So, whoever wakes up and does not know that the sun has risen, and prays believing that the time is still current, their prayer is valid even if it was after sunrise.