Connolly joined the action with a minute left, and, with his first touch of the ball, put Saints ahead in injury time.
Dan Harding's shot was saved by Shrews goalkeeper Chris Neal and the ball looped up invitingly for Connolly who reacted quickly to stab the ball home from close range.
A minute later Connolly fed Adam Lallana who curled a delightful right-foot shot around Neal to confirm the victory. Before that the Saints had struggled to break down a resolute Shrewsbury defence in what was a dour match.
Had Shrewsbury shown a little more of the ambition they showed just before the end of the first half they might have caused an upset.
First Dean Holden brought a sharp save from Saints' keeper Kelvin Davis and from the resulting corner Holden's header was smuggled off the line by a combination of Davis and winger Alex Chamberlain.