The closest the game came to a goal in a lifeless first half was when an El-Hadji Diouf corner caused problems for his former club, only for Ricardo Gardner to clear off the line.
Cisse scuffed a shot after being put through by Danny Collins, while Bolton's best chance was a weak effort from Johan Elmander.
Gardner had to clear off the goalline for a second time to deny a Jones header before Richardson's set-piece finally created the opener. He swung the ball in left-footed and Jones met it with a firm header.
Edwards then fired an 18-yard shot off the top of the crossbar after a neat passing exchange between Richardson and Jones as Sunderland stepped up a gear.
Cisse gave them some breathing space when he doubled the lead, volleying home Edwards' right-wing cross.
Bolton got themselves back in the match when Smolarek, who replaced the ineffective Kevin Davies, found himself in space wide on the right.
His looping centre caught out Fulop as it dropped over the goalkeeper and inside the far post.
It set up a frantic finish with Bolton throwing everything forward - including keeper Jussi Jaaskelainen - but Sunderland held on.
Sunderland manager Ricky Sbragia: "I am delighted that after a bad performance at Everton we got back on track. It is great to go through.
"I was reminded yesterday we hadn't qualified for the fourth round for three years. We changed it around a little bit and I felt that team we put out there could see us through it.
"I'm pleased with that and I'm pleased for the players."
Bolton boss Gary Megson: "I thought we set off fine first half. We closed the ball down well, we had opportunities but we didn't take one.
"It was nip and tuck and we mentioned at half-time the importance of the first goal and we gifted it to them.
"I don't take anything away from Sunderland because they have beaten us but in the first half the crowd were starting to get a bit upset."
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