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 after he had been set up by Sessegnon.
Source: PA
Source: PA