Considered among the best legspinners to come out of New Zealand, Alabaster played 21 Tests, taking 49 wickets between 1955 and 1972.

Alabaster was part of New Zealand's first four Test wins. He was on tours to India and Pakistan in 1955-56, England in 1958, South Africa in 1961-62 and West Indies in 1971-72.

He took 22 wickets in five Tests in the series in South Africa, helping New Zealand get their first series draw against them. He took seven wickets in the first match of the series as New Zealand narrowly lost by 30 runs, before returning his career-best match haul of 8 for 180 in New Zealand's first Test win abroad as they drew level 1-1 in the third Test.

Representing Otago in domestic cricket, in all he played 143 first-class games and claimed 500 scalps.

He played the first two Tests on the West Indies tour in 1971-72, bowling Garry Sobers for his only wicket of the series before an Achilles tendon injury cut his tour short. He retired from first-class cricket after that, which meant his first and last first-class games were both for New Zealand.