Born in Looe, Cornwall, he played for local non-league side Looe; a prolific centre forward, despite his slight frame he scored over 100 goals in a season, including ten in a single match. He was spotted by Plymouth Argyle, and signed for the Pilgrims as an amateur in 1926; he still worked as an auctioneer in his day job. By the start of the 1928-29 season, Bowden had impressed enough to win a first-team place, and was a regular goalscorer for the club for the next five seasons; Argyle were promoted in 1929-30 from the Third Division South to the Second Division. In all he scored 83 goals in 149 league matches for Argyle, more than a goal every other game.
Bowden signed for Herbert Chapman's Arsenal in March 1933 for £4,500, as the intended replacement for David Jack. Although he did not play enough games to win a First Division winners' medal that season, he was a regular for the next two seasons, as Arsenal won two more titles on the trot. Playing mainly as an inside forward, behind Ted Drake, he still scored his fair share despite his slight build in what was a very physical game at the time; his goalscoring included a haul of 13 in 1933-34, which made him Arsenal's joint-top score in the league, along with Cliff Bastin.
In 1934-35 Bowden scored 14 league goals and also won his first cap for England, against Wales on September 29, 1934. Two months later, Bowden was one of the seven Arsenal players who played for England against World Champions Italy in the "Battle of Highbury" match, which England won 3-2. In all Bowden represented his country six times, scoring once, against Wales in February 1936.
Bowden won an FA Cup winners' medal in 1935-36, but by then his ankle was causing him problems, limiting his appearances for the club that season and the next. By the start of the 1937-38 season he had seemingly bounced back, playing ten matches in the first two months of the season, but in a reshuffle of the Arsenal side he was sold to Second Division Newcastle United in November 1937 for £5,000. In all he played 138 matches for the Gunners, scoring 48 goals.
Bowden was a regular for Newcastle United for the next two years (although the club narrowly escaped relegation in his first season). With the outbreak of World War II, first-class football was suspended, and the 30-year-old Bowden decided to call it quits. He returned to Plymouth, where he ran a sports shop with his brother Austin. He died in 1998, aged 89.