Jordi Gomez found the back of the net after 77 minutes, Adam Johnson after 87 and Connor Wickham on 88.
But the scoreline was certainly not a fair reflection of the game, with Blues more than matching their top-flight opponents for the majority of the contest and creating plenty of chances of their own.
There were five changes to the Blues side that played out the 1-1 draw with Brentford at the weekend, with Colin Doyle, Neal Eardley, Paul Robinson, Mark Duffy and Stephen Gleeson all returning to the starting XI.
Blues made an extremely bright start and came close after just five minutes when Paul Caddis unleashed a shot from at least 25 yards, but Costel Pantilimon did well to push it behind for a corner at full stretch.
Having endured some early Blues pressure, Sunderland began to keep hold of the ball much more, but without really causing the hosts attacking problems.
Blues were tending to play more of their football on the counter-attack and Santiago Vergini was forced to chest David Cotterill’s dangerous cross behind for a corner.
With half an hour played Seb Larsson chipped an exquisite ball over the Blues defence for Jozy Altidore, but his Sunderland team-mate volleyed his shot high over the bar from just ten yards out.
Clark was forced into making a change after just 32 minutes when Eardley had to come off with a groin problem and he was replaced by Jonathan Spector.
Sunderland had a let off after Clayton Donaldson challenged Pantilimon for the ball and the keeper spilled it, but the Sunderland defence managed to scramble it away.
Minutes before the break Duffy slid a brilliant ball across the six-yard box, but under pressure Wes Thomas fired it wide from close range. Then less than 60 seconds later Cotterill hit the base of the post with a low drive, which deflected behind for a corner.
Half-time: Blues 0 Sunderland 0
Referee Darren Bond produced his first yellow card of the evening after 53 minutes when Thomas caught Liam Bridcutt.
A late challenge on Donaldson by Jones left the Blues striker writhing on the floor, but the referee just waved play on. This incensed both the Blues players and supporters and at same time raised the atmosphere around the ground.
Just after the hour mark Lee Novak became Blues’ second substitution when he was sent on for Duffy.
Sunderland won a free-kick in a great position just outside the Blues box, but Jordi Gomez fired it straight into the defensive wall. It subsequently bounced out to Bridcutt but his low shot was comfortably saved by Doyle.
At the other end Edgar tested Pantilimon with a pile-driver of an effort from 30 yards, before Gleeson fired over from distance.
Totally against the run of play Sunderland took the lead after 77 minutes through Gomez. The ball fell to him approximately 20 yards out and he let fly with a powerful low shot which evaded a diving Doyle and nestled in the back of the net.
With four minutes of normal time remaining Donaldson came desperately close to an equaliser with an overhead kick which went just the wrong side of the post.
But it was game over a minute later when Johnson slotted the ball home after Doyle had saved Altidore’s initial effort.
With Blues still reeling the Black Cats made it 3-0 a minute later after they broke forward and the ball found Wickham who made no mistake with his finish.