Essugo will earn Sporting an immediate €23 million (£19.5 million).
The deal also includes Geovany Quenda, who will stay at the club for another season on loan from Chelsea, bringing the total fee to a reported €74.4m (£62.8m).
Essugo was on loan from Sporting to Las Palmas in La Liga last season, where he made 27 appearances and scored one goal.
He was the youngest player in Sporting's history, making his first team debut in March 2021 against Vitória SC, aged 16 years and six days.
In doing so, he also broke Luis Figo's record for the youngest player to start for the club (aged 16 years, 11 months and 17 days).