Despite that quick start, Rovers found themselves 1-0 down soon after. Dale Stephens's ball over the top managed to find Church who, after controlling it on his chest, volleyed home.
Jordan Rhodes could have pulled Rovers level, but his header went straight into the hands of Ben Hamer.
At the other end, Marvin Sordell made space for himself but was unable to direct his shot past Jake Kean.
Hamer was again called into action, this time acrobatically turning Rhodes's volley round the post.
Rovers continued to create the better openings, but Best was unable to find the target when well placed.
Best had another chance for Rovers shortly after the restart but he failed to find the target with his header.
Stephens then connected with a Jordan Cousins cross but he directed his header at Kean.
Charlton substitute Yann Kermorgant tried an outrageous effort from his own half with his first touch, but saw the ball go wide.
Goalscorer Church volleyed wide for the Addicks, but it was the hosts who had the majority of the pressure, although the away side were able to see out the game.