Robinson joined Brighton from Manchester City in 1980 and scored 37 goals from 113 appearances.
A tweet from Brighton's official account said: "We are deeply saddened to learn of the passing of former striker Michael Robinson at the age of 61. Our thoughts are with all of his loved ones at this extremely difficult time."
He moved to Liverpool from Albion in 1983 for £250,000 and battled for a place in their starting line-up alongside the likes of Ian Rush and Kenny Dalglish.
Robinson also played for Preston North End, QPR and Spanish club Osasuna and won 24 caps for Republic of Ireland.
He won the league and European Cup with Liverpool and had a successful TV career in Spain after his retirement from playing.
A post on Robinson's official Twitter account said: "With tremendous sadness we inform you of Michael's death.
"It leaves us with a great emptiness, but also countless memories, full of the same love that you have shown him.
"We will be eternally grateful to you for making this man SO HAPPY, he never walked alone. Thank you."
Robinson announced in December 2018 he had been diagnosed with a malignant melanoma.
Barcelona tweeted: “A person who loved football and who knew how to explain it with knowledge and ingenuity."