Doohan and Bearman engaged in a thrilling battle, the pair coming close to contact on multiple occasions.
The Australian ultimately gained the upper hand, forcing his rival into a mistake to claim the final step of the podium.
A pre-race downpour saw the field follow the Safety Car for two formation laps.
It was a prudent move from officials as it was the first time the drivers had taken to the circuit in wet conditions throughout the weekend.
After three formation laps, the Safety Car was withdrawn and the race began with a rolling start.
Frederik Vesti headed the pack, which pulled out of his rooster tail to glimpse the opening corner.
In the pack, Brad Benevides found himself tipped into a spin approaching Aintree, pirouetting across the nose of a slowing Ayumu Iwasa.
The Safety Car was called for the stricken Benevides, who slid across the wet grass and into the barrier.
Iwasa had been slowing with his own malady, touring back to the pits where the DAMS crew set about bleeding the rear brakes for the Japanese driver.
Benevides had been helped on his way by Clement Novalak, the pair running well down the order at the time of the incident.
It was a brief interruption, and Vesti was left again in control of the race on Lap 4.
Oliver Bearman put a move on Theo Pourchaire at Village to claim third, having despatched with Enzo Fittipaldi before the Safety Car's emergence.
The Brit was on a charge, picking Isack Hadjar's pocket with a bold move at Stowe just half a lap later.
As he ended Lap 6, Bearman had a spin exiting Club, dropping back behind Hadjar.
His early pace had disappeared and instead of looking forward, was soon being harried by Pourchaire, the Frenchman making the move at Club at the end of the lap.
As that battle took place, Jack Doohan was duking it out with Enzo Fittipaldi for fifth, the Australian gaining the place.
Bearman soon had Doohan for company, the Virtuosi driver pulling off an audacious move around the outside of Village.
However, Bearman refused to concede and immediately fought back at The Loop, seizing the inside line on the right-hander to take the apex and, with it fourth place one more.
Ahead, Theo Pourchaire got the better of Hadjar.
Meanwhile, Doohan was still tucked up behind Bearman; clearly with pace in hand, there was no way through for the Australian.
He looked to have made the move at Stowe but appeared to have been forced off the road to remain fifth.
Officials noted the incident as Doohan complained about his rival's antics over the radio though nothing came of it.
Hadjar was also struggling for pace, dropping a place to Bearman at Copse as Doohan looked to take advantage.
The door remained shut, Hadjar defending at Stowe and again at Club which left Doohan exposed to attack from Fittipaldi.
Bearman remained second from Hadjar, Doohan and Fittipaldi until Brooklands on Lap 16.
Driving around the outside, he had the inside line and with it fourth place by the time they got to Luffield.
He promptly set off in pursuit of Bearman again, closing in on the Prema driver within half a lap.
Again, the Australian had superior pace but found getting by his rival no easier than it had been previously.
Doohan tried around the outside of Brooklands on Lap 17, as he had done with Hadjar, but the door was firmly closed once again.
At Stowe, Bearman again tried to hold Doohan wide, but could not.
The pair ran side by side into Vale only for a lock-up to send Bearman wide and hand the final podium place to Doohan with three laps remaining.
Fittipaldi and Hadjar were also able to get by Bearman as he recovered, the Brit noted for rejoining unsafely.
Out front, Vesti held a comfortable advantage over Pourchaire with Doohan third, Fittipaldi and then Hadjar, who headed a gaggle of cars that ran back to Roy Nissany in ninth.
The Dane held on to win by more than 13 seconds, with Pourchaire a lonely second from Doohan at the chequered flag.