Born in Heavitree near Exeter, Bastin started his career at Exeter City, making his debut for the club in 1928, at the age of 16. Despite only playing 17 games (and scoring 6 goals), his talent was evident enough that he was signed a year later for £2,000 by Herbert Chapman's Arsenal, who went on to become dominant force in English football in the 1930s.

Bastin would play the rest of his career at Arsenal, scoring 178 goals in 395 games; he was Arsenal's all-time top goalscorer until 1997, when his total was surpassed by Ian Wright. In 2005 Thierry Henry passed both Bastin and Wright's totals, thus meaning Bastin is currently (as of October 2005) Arsenal's third-top goalscorer of all time. Bastin's scoring feats are all the more remarkable considering he played on the left wing rather than as centre forward; the partnership he formed with Alex James was the source of many of his goals. Had his career not been interrupted by the Second World War it is likely that he would have scored many more goals for Arsenal.

Despite being so young, Bastin made an immediate impact and was a regular in the Arsenal side through the '30s, earning him the nickname "Boy Bastin". With the Gunners, Bastin won the FA Cup twice, in 1930 and 1936, and the Football League five times, in 1931, 1933, 1934, 1935 and 1938. Bastin also played for England 21 times, including a notorious match against Germany in Berlin in 1938, when the England team were ordered to give the Nazi salute before the match.

The Second World War intervened when Bastin was 27, thus cutting short what should have been the peak of his career. Bastin was excused military service, as he failed the army hearing test. Thus, during the war, he served as an ARP Warden, being stationed on top of Highbury stadium with Tom Whittaker. He also played matches in the war-time league (but, strangely, not internationals) to boost civilian morale. In 1941, Fascist Italy's propaganda broadcast on Rome Radio, contained a bizarre claim that Bastin had been captured in the Battle of Crete, and was being detained in Italy. The Italians were seemingly unaware that Bastin had played his entire career being almost entirely deaf.

Bastin had injured his right leg in the season before the war, which would go on to hamper his performances in wartime matches, and ultimately curtail his career. After the war was over, Bastin, by now in his thirties, would only play six more times before retiring in January 1947.

After retirement, Bastin returned to his native Exeter and ran a pub. He died in 1991 at the age of 79. A stand at St James Park, Exeter's home ground, is named in his honour.