The 23-year-old Irishman takes over from Italian Manuel Pascali, 33, who has moved to Cittadella in his homeland.
Connolly joined Killie from Crawley Town in 2014 and made 29 appearances for the Rugby Park side last season.
"To be captain of any football club is a real honour," Republic of Ireland Under-21 cap Connolly told the Kilmarnock website.
"There's a lot of hard work to be done this year being captain but we've got a good team behind me to help me out. There's a lot of leaders in there.
"I have captained under-age teams at Wolves and Bolton so I have always wanted to be a captain.
"That's the way I play every week - trying to lead by example."
Kilmarnock begin their Scottish Premiership campaign at home against Dundee on 1 August.
Kilmarnock have announced defender Mark Connolly as their new captain.