Performing fajr Salaah at home if one fears missing the Salaah

For whatever reason which Allah (Ta'ala) knows best I woke up for fajr salaah a bit later than usuall i.e. within permissible time but not enough to make it to the masjid as I felt I could miss the Jamaah and fall within an impermissible tome. I commenced with salaah and performed a short but not hesty salaah with Jamaah at home. I had completed salaah and tasbih* and dua by 04:35am- 20/12/2015-durban. Latest Jamaat commencment time in locality is at 04:25 (Masjid is Al-Hilal)- did my salaah fall within the permissible time and was my decision to perform salaah at home with jamaat at home valid on grounds that I could have missed salaah and permissible time completely had I driven to the masjid. Note that we follow the Shafi'i Madhab so the duration of salaah included dua e qunoot which was incumbent on me as the imam leading salaah.