The race was marred by a first-corner pile-up that involved 14 cars and took out polesitter Johnny Cecotto Jr (who sparked the chaos himself) as well as frontrunners Fabio Leimer, Jolyon Palmer, Robin Frijns, Marcus Ericsson and Alexander Rossi.
The event was red-flagged with nine of those cars unable to restart.
The restart took place under the safety car, with Mitch Evans up front in his Arden entry. Bird, in his repaired car after taking a big hit from Kevin Ceccon at the first start, was right on Evans's case from the green flag on lap two.
Evans had switched to the super-soft tyre during the red flag, and it was clearly a struggle to keep Bird, on the softs, at bay.
Evans did so, however, keeping the Briton behind him before pitting on lap 13. His strategy meant he had to change all four tyres to the soft compound, whereas when Bird pitted four laps later he only had to change his soft rears.
Evans's slower stop meant that Ceccon, who had a new front wing fitted during the red flag, managed to jump him after opting for an early pitstop.
But with Rene Binder taking a long first stint running second on the road, Ceccon was never able to glimpse a view of Bird, who scampered away to win by 22s as he added a GP2 Monaco win to last year's Formula Renault 3.5 triumph in Monte Carlo.
Bird's only scare came when he clipped the kerb exiting Portier soon after his stop, his car fishtailing perilously closely to the Armco.
Ceccon came under pressure for second from Evans in the closing stages, but pulled away as Evans needed to keep an eye on Felipe Nasr, who had detached himself from a fight with James Calado.
Nasr had one eye on the points race, however, closing the gap to leader Stefano Coletti to 15 with his fourth place.
Coletti finished sixth, ahead of Binder (a fine run after restarting 15th), while Adrian Quaife-Hobbs pulled off a last-lap move on Stephane Richelmi to grab reversed-grid pole position for Saturday's sprint race.