Having dispatched Wilfried Zaha 5-4 in the previous round, Sterling was involved in another thrilling, high-scoring contest with the Toffees’ midfielder.
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Sergio Aguero was the man who did the damage for the Manchester City winger, with the Argentine hitting a hat-trick after Kevin De Bruyne had opened the scoring.
In a close fought encounter, Gomes twice came from behind as the two teams went in level at half-time, but Sterling had too much for him after the break when Aguero netted twice to complete his treble.
Our No.7 will again face Merseyside opposition in the semi-final, where he will come against his England colleague, Liverpool's Trent Alexander-Arnold.
City fans in the UK can watch that clash on Sky Sports Main Event and Sky Sports Premier League at 15:00 BST on Saturday 25 April.
You can also watch all the action on the Premier League website and app as well as its YouTube, Facebook and Twitter channels.
The ePremier League Invitational prize fund is being donated to the #PlayersTogether initiative, which was created by Premier League players as a collective way of generating funds for the National Health Service and distributing them where they are needed most.