I'm assuming you're specifically referring to the traditional end of the Western Empire in 476? If that's the case, they didn't see it as a "fall" really. The Byzantines preserved a fiction of suzerainty over Italy by recognizing the de facto rulers as governors in the emperor's name.

In 472, there was a succession crisis in the west. The local foederati (the barbarian auxiliaries nominally under imperial control but were really the kingmakers by that time) supported one candidate, and the Eastern Emperor Leo I and then Zeno supported Julius Nepos. Nepos was installed with Eastern assistance and became Western Emperor.

Then Zeno was briefly deposed by a relation of his, Basiliskos, in 475. While Zeno was fighting to regain power, the foederati under Orestes launched a coup against Nepos, driving him from Ravenna into Dalmatia across the Adriatic. In his place Orestes installed his son, Romulus Augustus. But Orestes wasn't very popular and he (and his son) were soon deposed by Odoacer in 476.

By then Zeno regained the throne. Odoacer requested that Zeno appoint him governor of Italy in Zeno's name while Nepos requested that Zeno help him regain Italy. Zeno split the difference and instructed Odoacer to recognize Nepos as Emperor (but Nepos remained in Dalmatia, leaving Odoacer de facto ruler in Italy). Four years later, Nepos was assassinated and Odoacer added Dalmatia to his realm and transferred his allegiance to Zeno.

While not a direct answer, the beginning episodes of the History of Byzantium (started by a HoR listener) address how the Eastern Romans felt about the fall of the West -- worth a listen.