Jamie McIntyre levelled the tie on aggregate with a cool right-foot finish on 16 minutes and Stephen Curry scored twice after the break.
Jason Johnston pulled one back in the 85th minute to give the Comrades hope.
Institute will be at home in Tuesday's first leg against Carrick who finished one from bottom in the Premiership.
The second leg takes place in Carrickfergus three days later.
Institute have spent the last two seasons in the Championship, having been relegated from the Irish League's top division in April 2015.
Ballyclare, who had finished in third place - six points behind runners-up Institute - took a 1-0 lead into Friday's second leg at Drumahoe.
But that was wiped out when McIntyre - promoted to the starting time in place of Gareth Brown - ran on to a through ball and directed his right-foot finish past the exposed visiting keeper Paddy Flood.
Institute's Stephen Curry was denied by a superb save by Comrades keeper Flood who tipped away a shot which looked destined for the top corner.
Curry edged Institute into the overall lead on 62 minutes, lifting a first-time finish past Flood from Sammy Morrow's flick-on.
Nine minutes later Curry punished poor Ballyclare defending by netting his second to give Kevin Deery's men some breathing space.
Johnston's reply - after John Dobbin's shot came off the post - meant the Comrades needed to score just one more to go through on away goals, but Institute saw out the match to win 3-2 on aggregate.
Institute overturned a first-leg deficit to beat Ballyclare Comrades and will play in the promotion/relegation play-off against Carrick Rangers.