Ashurst was an apprentice with Liverpool in 1953, then released. He joined Sunderland in 1957, where he would go on to spend 14 years. He made his debut the following year against Ipswich Town and went on to make 458 appearances for the club; putting him second in the all time appearances list in Sunderland's history, and one of only two outfield players to top 400 appearances. He scored four Sunderland goals during his time at the club.
Ashurst was appointed manager of Newport County in 1978 following the departure of Colin Addison to West Bromwich Albion. Ashurst was manager from 1978 to 1982, the most successful period in the club's history. Newport were promoted to the Third Division in the 1979–80 season and won the Welsh Cup, entitling them to play in the 1980–81 season European Cup Winners' Cup, reaching the quarter finals. Ashurst was sacked by Newport County in February 1982 and Addison returned as team manager. The team, largely assembled by Ashurst, attained Newport County's highest post-war finish in the 1982–83 season, 4th in the Third Division, narrowly missing out on promotion.
His time as Sunderland manager was not successful, despite taking them to their first ever League Cup final. Performance in the league was poor and Sunderland were relegated from the first division. Ashurst was sacked in May 1985.
After returning to England, he was assistant manager of Blackpool, and then in September 1989 he returned for a second spell as manager of Cardiff City following the departure of Frank Burrows. He spent two years in Ninian Park before resigning in 1991 as the team struggled both on and off the pitch. His last managerial role was a one-year stay at Weymouth.
Since the mid-1990s, Ashurst has been heavily involved in an administrator's role at the Football Association specifically with regards to the Academy system.