Nick Markakis and Ryan Flaherty also connected for the Orioles.
Norris (5-5) struck out six and walked three while matching the longest outing of his career. Coming in, he had given up four or more runs in four of his past five starts.
Boston has lost six of its past seven games after a season-high seven game winning streak. The Red Sox have been shut out five times this season.
Jones had three hits, including his 150th career homer. Markakis extended his hitting streak to 14 games with two hits.
Boston right-hander Jake Peavy (1-4) allowed four runs and eight hits with seven strikeouts and one walk over seven innings. Peavy has given up three or more runs in seven consecutive starts and hasn't won since April 25 against Toronto.
After allowing a leadoff single to Brock Holt in the first, Norris retired 13 consecutive batters. Daniel Nava managed Boston's second hit with a two-out double in the fifth.
Jones gave the Orioles a lead in the first with his 10th homer.
Baltimore then increased the lead to 3-0 that same inning when Markakis hit a two-run shot off Peavy onto the flag court. Flaherty's solo shot in the seventh provided further cushion.
Tommy Hunter, who came off the disabled list Sunday, threw a scoreless ninth. The Orioles improved to 24-9 when scoring four or more runs.
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