Berg was born in Vancouver, British Columbia. He began karting in 1978 and transitioned to cars at the age of 20, racing in Formula Atlantic. In 1982 he won the Tasman Formula Pacific Series in Australia and New Zealand, a prestigious early-season championship that served as a testing ground for international talent. The following year he entered the British Formula Three series, but faced one of the most competitive grids in the championship's history: Senna and Brundle were fighting for the title, and Berg had to compete against both. He won one race — one in which Senna and Brundle opted out to score European Formula Three points — and finished fifth in the series overall. In 1984 he finished runner-up in British Formula Three behind Johnny Dumfries, recording eight second-place finishes. At the season's end he tested for both Arrows and Tyrrell and held discussions with Spirit and RAM Racing, but a Formula One drive did not materialise. He returned to Canada in 1985 to seek the funding necessary for an F1 campaign.
Berg's opportunity came in 1986 when he purchased a seat with the Osella team midway through the season. He took over the car vacated by Christian Danner, who had moved to Arrows to replace Marc Surer after Surer suffered career-ending injuries in the ADAC Hessen-Rallye. Berg competed in nine Grands Prix before his sponsorship ran out, forcing him to miss the Italian Grand Prix at Monza. Additional funding allowed him to return and complete the season. He did not score championship points during his time in Formula One. After the season he held advanced discussions with Larrousse and other teams about a 1987 drive, but the cancellation of the Canadian Grand Prix that year removed a key leverage point for attracting domestic sponsors, and he was unable to raise sufficient backing to continue in Formula One.
Berg remained active in motorsport across several disciplines. He drove a privateer BMW M3 in the Deutsche Tourenwagen Meisterschaft in 1991 and won the Mexican Formula Two championship in 1993. He also won the Indy Lights Panamericana title in 2001 as a driver-owner before retiring from competitive driving. He competed in sportscar racing and the Trans-Am Series during this period.
After retiring from driving, Berg founded Allen Berg Racing Schools, based primarily at WeatherTech Raceway Laguna Seca in Monterey, California. The school offers driving experience programmes in formula racing cars for people with little or no prior experience, as well as structured instructional programmes for drivers seeking professional careers. Several graduates of the school have gone on to race professionally.