Manager Mick McCarthy's decision to make nine changes to the side held to a last-gasp draw by West Brom at the weekend was in danger of back-firing as the League Two side proved more than a match.
Sunderland were reduced to 10 men in stoppage time when 18-year-old Dan Smith, making his debut for his home town club, was sent off for a second bookable offence.
But Anthony Le Tallec, on loan from Liverpool, spared the blushes when he lashed home Liam Lawrence's right-wing cross in the second minute of extra time and substitute Stephen Elliott hit the post as well as having another effort well saved.
Cheltenham were far from overawed in the first ever meeting of the two clubs, competing fiercely in midfield to deny the home side time and space to create openings.
Jonathan Stead and Le Tallec could make little impression on a sound visiting defence, though the former came closest to breaking the deadlock in a lacklustre first half.
Le Tallec split the Cheltenham defence with a majestic 30-yard pass only for Stead's first-time effort to produce a fine diving save from Shane Higgs.
Nyron Nosworthy, one of the two survivors from the weekend, centred for Le Tallec to head over the bar, but the visitors soon showed their attacking prowess to test rookie goalkeeper Ben Alnwick.
Danny Collins was pressed into conceding the first corner and then a free-kick against Carl Robinson, taken by Brian Wilson, was well saved by the youngster deputising for an out-of-sorts Kelvin Davis.
The Black Cats caused their own problems by being caught in possession, Christian Bassila giving Grant McCann the opportunity to let fly from 35 yards but the shot failed to trouble Alnwick.
Jamie Victory failed to hit the target with a header from McCann's corner and Stead fared no better at the other end from Smith's pin-point centre.
Sunderland were on the rack in the early stages of the second half and twice within a minute could have fallen behind.
Kayode Odejayi pulled the ball back only for McCann to shoot wide from a great position.
McCann turned provider immediately, finding Ashley Vincent whose dangerous cross was turned behind by the out-stetched foot of Alnwick.
But Sunderland should have gone ahead in the 56th minute. Lawrence found Le Tallec at the far post with a deep centre but from only a few yards out the on loan striker from Liverpool headed wide.
McCarthy's response was to haul off Matt Piper and Stead in favour of Andy Welsh and Elliott and the double substitution almost paid instant dividends.
Elliott latched on to Lawrence's through ball and chipped over Higgs only for Wilson to hook clear just short of the line.
Bassila and Le Tallec failed to direct headers on target as the Black Cats gradually stepped up a gear but Cheltenham replied by sending on Damian Spencer and Jeremy Gill in place of Odejayi and John Melligan as both teams sought a winner.
But Sunderland overcame their numerical disadvantage to finally win through to the delight of a sparse crowd of just under 12,000.