The Foxes' persistent pressure early on had paid off in the 14th minute with a finely crafted goal when Connolly dummied Jason Wilcox's cross to allow Mark De Vries the time to control the ball before driving beyond Steve Simonsen from seven yards out.
Stoke, who had started brightly, had to wait until the 38th minute for their first worthwhile effort when Darel Russell shot narrowly wide from 16 yards.
And the visitors squandered a great chance to level after the restart when Junior's throughball found Mamady Sidibe with only the keeper to beat but he sliced wide from eight yards.
Momo Sylla then shot high and wide for Leicester and Luke Chadwick dragged a shot just wide for the visitors.
On the hour mark Stoke were reduced to 10 men when Junior lashed out at Johannes Gudjonsson, catching the Icelander with his boot. Referee Paul Taylor consulted his assistant before showing a straight red.
But undeterred, Stoke rallied to equalise on 64 minutes when David Brammer's far post corner was headed back across goal by Carl Hoefkens for Marlon Broomes to hook home from eight yards.
But their delight was shattered with Leicester restoring their lead after 66 minutes.
Gudjohnson drove a long free-kick into the penalty area and when the ball accidently struck Connolly, the striker reacted smartly by spinning and firing beyond Simonsen from 10 yards.
Leicester appeared to have applied the killer blow on 82 minutes when Gareth Williams broke forward and found Connolly who beat the keeper with a deft shot into the far corner.
However, Stoke gave themselves a fighting chance with just one minute left when slack Leicester defending allowed Chadwick to find John Halls who slotted home from 10 yards.
But Connolly put the game beyond doubt in time added on when he seized on Brammer's weak back pass to slot the ball beyond Simonsen from close range.