Bryant took Jarred Cosart for a two-run homer in the first inning and went long again with the bases loaded in the second to power his side to a 7-2 success.
The Cubs remain 8.5 games behind National League Central leaders the St Louis Cardinals, who ended an uncharacteristic four-game losing streak with a 2-1 win over the San Diego Padres.
The Pittsburgh Pirates beat the Cleveland Indians 1-0 and the Milwaukee Brewers won 7-3 against the Cincinnati Reds.
The Washington Nationals moved 4.5 games clear in the NL East, beating the San Francisco Giants 9-3 while their nearest pursuers the New York Mets lost 4-3 to the Los Angeles Dodgers. Those results also saw the Dodgers move four games clear of the Giants in the NL West.
The Atlanta Braves won 9-5 against the Philadelphia Phillies while the Arizona Diamondbacks beat the Colorado Rockies 7-3.
The New York Yankees moved two games clear in the American League East with a 3-2 win over divisional rivals the Tampa Bay Rays as Brad Boxberger’s fielding error in the bottom of the ninth saw Rays hitter Steven Souza Jr’s two-run homer go to waste.
The second-placed Baltimore Orioles lost 3-2 to the Chicago White Sox but remain a game ahead of the Rays and the Toronto Blue Jays, who went down 8-3 to the Detroit Tigers after homers from JD Martinez and Chris Colabello.
The Boston Red Sox complete the AL East just seven games off the summit after beating AL West leaders the Houston Astros 6-1 behind the hitting of Xander Bogaerts and Mookie Betts.
The Los Angeles Angels closed within four games of Houston after a 13-0 rout of the Texas Rangers, CJ Cron with six RBIs including a three-run homer. The Oakland Athletics beat the Seattle Mariners 2-0.
Detroit lie third in the AL Central behind the Minnesota Twins and Kansas City, who met at Kauffman Stadium with the Twins coming out on top 5-3 to narrow the gap to 3.5 games. Joe Mauer hit a two-run homer.