The original line-up consisted of Dave "Biffo" Beech (vocals and drums), Clive Scott (keyboards and vocals), Barrie Bernard (bass guitar), Tony Campbell (guitar), Tony Britnell (saxophone) and Kevin "Beppy" Mahon (saxophone). Biffo left in the first year, and was replaced by Des Dyer (drums), who took over lead vocals.
Scott was previously in Scott and The Antarctics, Dyer had played in both The Surfcyders and Clockwork Shoppe, Campbell, Mahon & Beech had worked together previously in The Mighty Avengers, Bernard had been a member of Pinkerton's Assorted Colours, whilst Britnel had worked with The Fortunes.
The group gives credit to their producer and manager, Chas Peate, for inspiration to get songs completed. As Jigsaw toured a lot, Scott and Dyer would only write snippets of songs separately in their spare time, but, as Dyer said in an interview in the 1990s, "it was Chas who would say we're recording an album soon, you'll need 10 songs by then". The two would then combine their work together for the planned recording sessions.
As songwriters the duo had a hit song with "Who Do You Think You Are?", which was covered by the UK band Candlewick Green in 1974. The song was further covered in the U.S. by Bo Donaldson and the Heywoods (this version made the U.S. charts), and again in 1993 by Saint Etienne. The song was also on Jigsaw's 1974 album I’ve Seen The Film, I’ve Read The Book. However, the band's own single from the album "You’re Not The Only Girl" was not a chart success and they were soon dropped by their recording label, BASF.
Wanting to carry on, the group continued to make music and signed a new deal with independent UK label Splash, which was founded by Chas Peate himself. Their first single for their new label, entitled "Sky High", was recorded for the film The Man from Hong Kong, a 1975 martial-arts action movie starring George Lazenby. Splash pressed a couple of hundred promotional copies and reaction to the song was good, especially from Radio Luxembourg which began playing the song with high rotation during its nightly English pop music broadcasts to the UK. "Sky High" subsequently made the UK Top 10 in November 1975. Chelsea Records released it in the U.S. (where Splash had no pressing plant) and it was a hit there, going to #3. It charted all over the world and was #1 in Japan. A famous Mexican wrestler called Mil Mascaras used "Sky High" as his theme tune. He would walk to the ring with the song playing in the background. His popularity in Japan helped "Sky High" to become a hit twice over in that country in consecutive years.
Their album also entitled Sky High sold very well too, especially in Japan, and the next single "Love Fire" was another hit, although not in the United Kingdom. Nevertheless, "Love Fire" was followed by two new albums, one for the British market called Pieces of Magic and another for the American market, the eponymous Jigsaw.
In 1977, the band recorded their album Journey Into Space in Los Angeles. Years later, Clive Scott revealed in an interview that the record label (Twentieth Century Fox) paid for the band to fly out to make the album, paid for their accommodation, and gave them cash advances (or, as Scott called it, “pocket money”) to live off for three months while making the record. The album was produced by Rick Jerrard who was the producer of Jose Feliciano and the person responsible for putting together Elvis Presley's Las Vegas band. However, the album was never released, much to the frustration of the group.
Jigsaw returned to Britain and scored another hit with the single "If I Have To Go Away", after which they recorded ten original songs for the film Home Before Midnight.
A little-known fact is that Bernard left the band in 1978 and was replaced by Jon Fox. Campbell left the band in 1979 and was replaced by Rod Godwin. The band, as a unit, finally ceased touring in 1981 but Scott and Dyer continued to record using Tim Stone on guitar and Pete Dennis on bass. This new group lineup recorded one album for Elektra. The album, a disco influenced set (that was never released in the United Kingdom) was issued in the United States the following year. A final single under the Jigsaw name was released by Splash in early 1983 called "Love Isn't At Home".