Sordell slid home a 64th-minute penalty and wrapped up victory nine minutes later as Dougie Freedman's men made sure of victory 10 days after blowing a two-goal lead in their Reebok Stadium clash.
Martin O'Neill had made his intentions clear by naming a strong side for the replay in contrast to Freedman, whose own slender play-off aspirations prompted him to make six changes from the weekend draw with Millwall.
There was little in the first half to suggest the Black Cats would not ultimately cruise home, with Adam Johnson twice denied by superb saves from Andrew Lonergan while the Trotters offered next to nothing in response.
The distinctly chilly Wearside atmosphere was barely warmed by the first half action, the best of which came from Johnson, whose rising 21st-minute shot was punched over the bar by Lonergan.
Johnson's resulting corner flashed dangerously across the face of goal and Stephane Sessegnon screwed a low shot just wide of Lonergan's right-hand post in the 25th minute as the home side began to look more dangerous.
With Bolton offering very little in the way of threat, Sunderland continued to have a smattering of chances, David Vaughan the latest to try his luck with a long-range drive which clattered Zat Knight.
Johnson once again came close to breaking the deadlock five minutes from his break when Lonergan tipped his low drive round the post and from the resulting corner Titus Bramble dragged a great chance wide.
Making only his second start of the season, Connor Wickham headed a cross from James McClean into Lonergan's arms, then Craig Gardner came close with a free-kick in the first action of the second period.
Until that point Sunderland had looked entirely dominant with Bolton failing to muster anything of note in the home side's half and Freedman cutting a frustrated figure in front of the away dugout.
But it was the home fans who were left unhappy as Bolton seized the initiative, almost grabbing the lead in the 55th minute when a dismal clearance by Simon Mignolet let in Sordell, who seemed as surprised as anyone and fluffed his chance.
Sordell and Chris Eagles had half-chances before Bolton's opener arrived courtesy of a careless challenge by Jack Colback on Darren Pratley just inside the box, allowing Sordell to convert from the penalty spot.
Despite a double Sunderland substitution, the visitors increased their lead 10 minutes later when Eagles threaded a fine ball to Tyrone Mears, who squared for Sordell to force his second under Mignolet.
There was more good news for Bolton's small band of fans before the restart as Stuart Holden came off the bench to make only his second appearance in almost two years due to an horrific knee injury.
Despite substitute Sebastian Larsson coming close with a late free-kick, there seldom seemed much danger of Bolton squandering a two-goal lead for the second time, as the Trotters cruised into round four.