The visitors went ahead in first-half stoppage-time when Bolo Zenden skipped around goalkeeper Jussi Jaaskelainen to score his first goal of the season after being set up by Stephane Sessegnon.
Sunderland almost extended their lead a minute into the second half when Sessegnon forced a good save out of Jaaskelainen but Bolton responded in the 50th minute with Daniel Sturridge firing a shot narrowly wide of the post and six minutes later Matt Taylor shot over.
Midway through the second half, Lee Chung-yong failed to find the target from close range, then 10 minutes from time substitute Rodrigo carved a path through Sunderland's defence but saw his shot beaten away by Simon Mignolet.
Bolton appeared to have salvaged a point in the 87th minute when Rodrigo delivered a superb ball to the back post and Ivan Klasnic, who had also stepped off the bench, headed into the far corner.
However, after Rodrigo had almost scored a stoppage-time winner for the hosts, Muntari made the decisive breakthrough for Sunderland, forcing the ball home despite the efforts of Bolton defender Zat Knight to keep it out after Sessegnon had created the opening.
Source: PA
Source: PA