Is my action correct in that I performed ghusl for Fajr prayer just before sunrise, which led to the prayer time expiring by a few minutes?

What you did first by purifying yourself is correct, because the preponderant opinion is that purification takes precedence over catching up on the time with tayammum (definition). However, ablution (wudu') does not substitute for ritual bathing (ghusl (definition)). The disagreement here is whether one should perform tayammum and pray, or perform ghusl and pray, regardless of whether the time for prayer is caught up or not.