Does the Dawn Prayer become an obligation to make up (qada') if one wakes up one minute before sunrise after performing ablution?

What you did by performing ablution after waking up, fearing the time would expire, is the obligatory action according to the majority of scholars.

Scholars differed on what constitutes performing the prayer within its prescribed time: 1. First opinion: The time is met by catching the opening takbir within the time, and according to this, your prayer is considered performed on time, not a make-up. 2. Second opinion: The time is not met except by catching one rak'ah.

According to both opinions, your prayer is valid and sufficient, and the obligation has been fulfilled.