Graham Potter made 11 changes to the side which started Brighton's last Premier League game.
But it didn't have much impact as the in-form Seagulls sealed a 2-0 victory at the AMEX.
Both goals came from Connolly, the first was a product of some fine work from Steven Alzate and Alexis Mac Allister which allowed the Republic of Ireland international to fire home.
Connolly doubled his tally five minutes later as he cut in from the right and fired a low shot into the bottom corner past Steven Benda.
Albion will now travel to Leicester for their fourth-round clash.
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