The K-Mets, winners of six straight and 14 of their last 16, are two games back from West Division leader Greeneville.
Yeixon Ruiz, Victor Cruzado and Joe Tuschak led the Kingsport offense with three hits apiece. Richie Rodriguez and Bradley Marquez added two each. Cruzado and Rodriguez both knocked in two runs.
The Mets (22-14) fell behind 2-0 before pulling a run back in the fourth and taking control in the fifth with four more to go up 5-2.
Kingsport pitchers held Bristol hitless after the third inning.
Robert Whalen (2-1) went six innings, giving up three runs (one earned) on three hits and two walks while striking out seven.
The K-Mets said goodbye to Marquez, who is returning to Texas Tech this week to participate in football training camp. He is a junior wide receiver for the Red Raiders.
Kingsport, which leads Elizabethton by 3½ games for a playoff spot, starts a three-game series at the Twins on Monday at 7 p.m.