The hosts took the lead on the quarter-hour mark through record signing Jurgen Locadia, who was making his debut for the Seagulls following his move from PSV Eindhoven in January.
Locadia redeemed himself when he swept home Anthony Knockaert's low cross to give the Premier League outfit the breakthrough.
The Seagulls struck again just after the hour mark, with Bruno's cross from the right corner of the box met by Leonardo Ulloa for a header into the far corner from around 12 yards out and a first Brighton goal for the Argentine since 2014.
Clarke-Harris rattled the woodwork again shortly after the restart and was eventually rewarded with a goal in the 77th minute, driving home after a poor Goldson clearance from goalkeeper Lee Burge.
Coventry could not find a second to give their opponents a nervy finish, however, and the final whistle saw their cup run come to an end as the south coast outfit reached the last eight of the competition since 1986.
BRIGHTON & HOVE ALBION (4-4-2): Krul (Maenpaa 45'); Bruno, Goldson, Hunemeier, Suttner; Knockaert, Kayal, Stephens (Propper 45'), March; Locadia (Baldock 77'), Ulloa
COVENTRY CITY (4-4-1-1): Burge, Grimmer (McDonald 57'), Willis, Hyam, Haynes; Bayliss, Kelly, Shipley, Barrett (Biamou 79'); Clarke-Harris; McNulty (Ponticelli 74')