Birkenshaw played a single County Championship match for Yorkshire at the age of 17 in 1958,[2] taking the wicket of Jim Parks in both innings, but did not make another first-class appearance until 1959.
He earned a place in the history books by playing (exclusively as a batsman) in the pioneering Midlands Knock-Out Cup limited-overs tournament in 1962.
The years between 1969 and 1972 saw Birkenshaw record solid statistics, taking 69, 63, 89 and 90 wickets in successive seasons, and in 1972 he achieved his career-best bowling of 8–94 against Somerset.
Such performances attracted the interest of the England selectors, who had previously stuck by Ray Illingworth as their first-choice off-spinner, and Birkenshaw was picked to go to India and Pakistan in 1972/73.
Despite another decent domestic season in 1973, Birkenshaw was omitted from the England side for both series that summer (against New Zealand and West Indies), the 41-year-old Illingworth again being the barrier to his selection.
He played on for Leicestershire for a number of years, helping them win the County Championship in 1975, making his fourth and final first-class hundred, and recording his best bowling figures in one-day cricket, taking 5–20 against Essex in the Sunday League.
In 1976, Birkenshaw made his only one-day hundred, but his unbeaten 101 against Hampshire in the second round of the Gillette Cup failed to win the game, Leicestershire falling short of their target by three runs.