The accident happened on Saturday near the Swedish capital Stockholm.
Local police said divers recovered the bodies of the victims — aged between 20 and 35 — after the vehicle plunged more than 25m (82ft) through a gap in the bridge which had opened to let a boat pass underneath.
The bridge, at the Saltskogs junction between the E4 and the E20 motorways, has a middle section that rises directly upwards without tilting, leaving a gap that the Nissan Qashqai drove into, Swedish police said.
A barrier, 50m before the opening, has flashing lights and signs warning there is a bridge opening, the officer handling the case said.
Inspector Martin Bergholm said: “For some reason, the car drove through the barriers and crashed down into the canal.”
Drivers of other cars were waiting behind the barrier but it is not yet known whether a boat had already passed or was waiting to pass.
Mr Bergholm added: “The witnesses just saw a car beside them and kind of disappear.”
The Warrington-based band members were Kris Leonard, River Reeves, Tomas Lowe, and Jack Dakin.
Tributes have been paid on social media for the four-piece band who were from Warrington, Cheshire.
Manager Craig Tarry is understood to be the fifth person in the car.
The British Foreign Office said it was in contact with the local authorities and supporting the families of the victims.
Band The Enemy wrote on social media: “RIP Viola Beach and their manager. Such incredibly tragic and sad news. Thought with the friends and families of all involved x.”
The up-and-coming band were on tour and due to play a string of dates at venues across the UK.
They were due to play a gig in Guildford in Surrey on Saturday, and were part of the line-up for Swedish music festival Where is the Music? on Friday.
British indie rock band Viola Beach killed in car accident https://t.co/idJQf3jluc— Rolling Stone (@RollingStone) February 14, 2016