Goddard was born in Smethwick, West Midlands, England, is the daughter of the British producer David Goddard and attended Farnham Girls' Grammar School, before he moved the family to Australia when she was 15 upon his appointment as Head of Drama at the Australian Broadcasting Corporation.

Goddard made an early television appearance in Australia on Skippy the Bush Kangaroo. After returning to the UK in 1969 as an adult, she appeared as Victoria in Take Three Girls (1969), and then its sequel Take Three Women (1982). She also had a supporting role in the 1972 movie Ooh… You Are Awful. Her career breakthrough was as April in The Brothers (1972–76), which also featured her first husband, Colin Baker. She appeared as Jocelyn in National Pelmet, the Series 2 opener of critically acclaimed drama Minder.

A comedy role in a sitcom followed, with a part in Pig in the Middle (1980), she was one of the 'explorers' who were evaporated in a (now missing) episode of the BBC science fiction quiz programme The Adventure Game (1980), played a space pirate in the 1983 Doctor Who story Terminus, and appeared in Roll Over Beethoven (1985). Goddard appeared in Woof!, a Children's ITV programme first aired in 1989. This long-running series told the adventures of a schoolboy who turned into a dog at any given time; his efforts to conceal his secret from his parents were aided by his teacher Mrs Jessop, played by Goddard. Her husband, producer and director David Cobham, created this series and showcased Goddard's talents to a whole new generation of fans. She had earlier appeared in the TV adaptation of Brendon Chase, also produced and directed by Cobham.

For many years, she was also the female team leader on long-running quiz/panel show Give Us A Clue, replacing another actress Una Stubbs in the role.

She appeared as Laurel Manasotti, who had an affair with Jimmy Mulville's character Donald, in the ITV sitcom That's Love.

She later had a recurring role as Philippa Vale in Bergerac. Her most recent recurring television role was alongside Dawn French and Catherine Tate in Wild West. In 2007 she appeared in the Midsomer Murders episode "A Picture of Innocence", reuniting her with Bergerac star John Nettles.