The Premier League outfit started the stronger and almost took the lead on 38 minutes as a free-kick from Adam Johnson was heading for the top corner until Andy Lonergan tipped the ball over the bar.
The Bolton keeper also saved from Connor Wickham just before half-time as the teams went into the break level, and then Sunderland enjoyed the first close chance of the second half as Craig Gardner hit a free-kick from long range just over the bar.
On 53 minutes, Marvin Sordell almost gave the visitors the lead as he beat the offside trap and Simon Mignolet had to turn it away.
But then Sordell put Bolton ahead on 64 minutes, Jack Colback gave away a penalty for a foul on Darren Pratley and the strike scored with a powerful effort to give the Championship outfit the lead.
Only nine minutes later and Sordell had got a second as he benefitted from poor defending to prod home inside the area following a Bolton corner, and it proved to be enough for Dougie Freedman’s men as they booked their progression to round four.
Ipswich striker Joe Garner will not play again this season after it emerged he suffered a fractured skull against Sheffield United last week.