The Benin international edged the Barclays Premier League side past their npower Championship neighbours, but they were taken all the way as the Teessiders came within eight minutes of a penalty shoot-out at the end of a tense replay.
Sessegnon's goal booked the Wearsiders a last 16 clash - their first in eight years - with Arsenal at the Stadium of Light a week on Saturday, seven days after the Gunners head north in the league.
The Black Cats took a 42nd-minute lead through Jack Colback's fine left-foot strike, but Lukas Jutkiewicz first Boro goal took the tie beyond 90 minutes.
However, Sessegnon ensured there would be no need for the most dramatic of finishes when he thumped home a late winner to main his side's fine run of form.
The locals among a bumper crowd of 26,707 roared the home side on throughout, but Premier League quality eventually told - just.
Boro went into the game without five of the men who had started the first encounter at the Stadium of Light - goalkeeper Danny Coyne, midfielders Barry Robson and Faris Haroun and strikers Scott McDonald and Marvin Emnes - to face a side which had lost just two of the 12 previous games they had played under O'Neill.
Sebastian Larsson tested keeper Jason Steele with a curling fourth-minute free-kick as the Wearsiders pushed Boro deep into their own half.
Mowbray had tasked right-back Justin Hoyte and defensive midfielder Tony McMahon ahead of him to keep man-of-the-moment James McClean quiet, and although the Irishman showed flashes of his best form, they were largely successful.
The Teessiders gradually eased themselves into the game and as the half wore on, it was they who created the better chances.
Sunderland keeper Simon Mignolet needed two attempts to grab former Black Cat Julio Arca's dipping 13th-minute effort, and John O'Shea did well to cut out a long-range effort from McMahon with Jutkiewicz lurking menacingly behind him three minutes later.
The Championship side, who had taken the lead at the Stadium of Light, might have done so once again at the Riverside Stadium when Jutkiewicz flicked on McMahon's 31st-minute free-kick and defender Seb Hines stabbed an instinctive shot towards goal which Mignolet tipped over at full-stretch.
But just as Boro sensed they were gaining the upper hand, the visitors hit back in devastating style.
Fraizer Campbell, back in the starting line-up after being rested at the Britannia Stadium did well to lay off a 42nd-minute long ball into the path of Colback, who controlled instantly before smashing a left-foot volley home off the underside of the bar.
Tony Mowbray's men might have been back in the game almost immediately, but Phil Bardsley cleared McMahon's skidding drive off the line, and Mignolet had to get down well to palm away 19-year-old striker Curtis Main's 44th-minute effort.
Sunderland returned knowing a second goal might effectively seal their passage, and they might have got it within 10 minutes when defender Michael Turner headed McClean's near-post cross over.
Boro who dragged themselves back into the tie with 57 minutes gone. Jutkiewicz was first to react to a loose ball inside the penalty area, then planting a firm right-foot shot past Mignolet to level.
The Sunderland keeper was relieved to clutch McMahon's volley to his midriff four minutes later, before opposite number Steele was tested much more rigorously by Craig Gardner's swerving right-foot strike.
Full-back Joe Bennett did just enough to prevent Larsson converting Sessegnon's 68th-minute cross, but Boro were grateful for the woodwork when Bardsley's deflected piledriver came back off the post four minutes later.
Steele beat away another well-struck Bardsley effort with 11 minutes left on the clock and the game went into extra-time.
Boro looked the more threatening as the extra 30 minutes started, but their fate was sealed in the 112th minute when Sessegnon drilled home from 15 yards after the ball broke to him to snatch victory for Sunderland.