Constantine wanted his city to copy Rome in nearly every respect. Rome had 7 hills so Constantinople would too- even if it took some creative counting to reach seven on the Bosporus. He naturally intended for a Senate as well- not to replace the Roman one but to mirror it. But despite Constantine’s best effort, Constantinople’s version of the Senate was much less respected and originally functioned as a kind of city magistrate. As the importance of the city grew so did its Senate’s prestige.
Although power shifted decisively east in the 4th and 5th centuries, the two bodies- Roman and Constantinopolitan- existed side by side for a surprisingly long time. In the West the Senate outlasted the empire by several centuries. They continued to meet and pass legislation (mostly pertaining to city affairs) after the last emperor was put out to pasture in 476. Theodoric consulted them when he took possession of the city and Justinian officially restored some of their ancient privileges in the sixth century. There are records of them acclaiming eastern emperors until the seventh century and although they probably stopped meeting as a group sometime after that, senatorial families were still powerful in Medieval Italy. (The Orsini family for example claimed to be descended from the Julio-Claudians- a member of their clan was raised to the papal throne as late as the 18th century)
In the East, the Senate remained as a functioning body until the 13th century. Their last recorded act was raising the unfortunate Nicholaus Kanabus as emperor during the 4th Crusade. Their power had been declining for centuries (as detailed here) and their titles were empty, but there were Senators defending Constantinople’s walls on the morning of May 29, 1453.