Perone, a Democrat representing the 137th District, received 502 votes, defeating challenger David Watts, who received 284 votes.
Morris, a Democrat representing the 140th District, received 599 votes, defeating challenger Warren Peña, who received 213 votes.
Both Perone and Morris will appear on the ballot in November's election. No Republican candidates have yet been announced.
Incumbent Norwalk Republican Registrar of Voters Karen Doyle Lyons also defeated her challenger John Federici. Lyons received 852 votes to Federici's 710 votes.
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