In a statement, talent agency DNA Music Ltd said Hawkshaw died in the early hours of Saturday morning after being admitted to hospital with pneumonia earlier this week.
He had spent the last few months recovering from a fourth stroke which he suffered in July.
The statement called Hawkshaw "one of the most sampled musicians in the world", adding: "Alan was behind such incredible theme tunes as Grange Hill, Channel 4 News, Countdown, Dave Allen at Large, and Arthur C Clarke's Mysterious World to name but a few.
"He played on a staggering 7,000 prominent recording artist music sessions including Olivia Newton-John, Serge Gainsbourg, Tom Jones, Barbara Streisand (and) David Bowie.
"We can guarantee if you don't know Alan's name - you will know his incomparable music."
Hawkshaw's German wife Christiane called his death "heartbreaking".
She said: "We spent the last few hours gazing at each other with love, holding hands, no need for words. I told him he and I were forever, and even though he and I had been unable to speak for the past two months, he managed a few 'forevers' and I knew he was at peace."
The Leeds-born musician wrote the music for "more than 35 films and countless television programmes", according to his website.
Hawkshaw began his career in the 1960s as a member of the band Emile Ford & The Checkmates, who toured with The Rolling Stones.
In the 1970s he worked as Newton-John's musical director and arranger/pianist, and received an Academy Award for best arrangement for the song I Honestly Love You.
In 2004, he set up The Alan Hawkshaw Foundation in association with the Performing Rights Society.
The scholarship programme provides financial support to gifted young musicians at the Leeds College of Music, now the Leeds Conservatoire, and the National Film & Television School.
After leaving school he did an apprenticeship at Leeds printing firm Knight & Forster on Water Lane, before leaving in 1959 when a band he played part-time gigs with turned professional and invited him to tour with them as piano player.