First will take on the route from Ashton Vale to Temple Meads once it opens towards the end of 2017.
MetroBus said the operator would not get public money to run the service and would "take the business risk".
Managing director of First Bristol James Freeman said the scheme marked "the start of an exciting new era for public transport in Bristol".
The rapid transit scheme was devised by the West of England Partnership - an alliance between the four councils in Bristol, Bath, South Gloucestershire and North Somerset.
It involves three routes - Ashton Vale to Temple Meads; Cribbs Causeway via Emersons Green to the city centre; and the South Bristol link between the A370 Long Ashton bypass and Hengrove Park.
It will provide an express bus service on 31 miles (50km) of segregated lanes.
The Ashton Vale to Temple Meads route is close to completion, with a new bridge over the Bristol to Portishead railway line and the Â£3.8m refurbishment of Ashton Avenue Bridge now complete.
Councillor Asher Craig, from Bristol City Council, said: "I am very much looking forward to MetroBus services starting.
"These measures and new smart ticketing will generate a noticeable uplift of quality and reliability to the public transport network."
First Bristol has been named as the first bus operator to run the Â£216m MetroBus transport scheme in Bristol.