The game was tied at 2-2 in the bottom of the ninth when Norris went deep to send in Cory Spangenberg, Alexi Amarista and Yangervis Solarte.
Josh Hamilton scored his first home runs since returning to Texas as the Rangers beat the Boston Red Sox 7-4.
Hamilton was traded back to the Rangers, where he spent five seasons from 2008 to 2012, by the Los Angeles Angels in April and made his first major impact as he cleared the fence twice.
Chad Bettis pitched seven innings without a hit before giving up two in the eighth as the Colorado Rockies beat the Philadelphia Phillies 4-1.
Bettis was in line for a perfect game through six innings but had to make do with the win that took the Rockies to 20-26 on the season.
Alcides Escobar, Alex Gordon and Salvador Perez all hit homers as the American League Central leading Kansas City Royals beat the Chicago Cubs 8-4.
City neighbours the White Sox had more luck as they beat the Houston Astros 6-3 in 11 innings while J.J. Hardy’s ninth-inning single gave the Baltimore Orioles a 2-1 win over the Tampa Bay Rays.
The Seattle Mariners were also 2-1 winners against the Cleveland Indians while the New York Mets lost 4-3 to the Miami Marlins.
The New York Yankees were also beaten as they went down 6-2 to the Oakland Athletics and the Cincinnati Reds beat the Washington Nationals 5-2.
The Los Angeles Angels kept the Detroit Tigers off the board in a 2-0 win but neighbours the Dodgers were held scoreless themselves, beaten 3-0 by the St Louis Cardinals.
In the day’s other games, the Arizona Diamondbacks beat the Milwaukee Brewers 7-5, the Toronto Blue Jays were 6-4 victors over the Minnesota Twins and the San Francisco Giants beat the Atlanta Braves 4-2.