Brees completed 18 throws in 23 attempts for 371 yards.
Three of the scoring passes came in a 21-point second quarter and the Saints were on their way to their 11th consecutive win this season.
New Orleans and Indianapolis, also 11-0, are the only unbeaten teams in the NFL.
Laurence Maroney scored touchdowns for New England (7-4) on runs of 4 and 2 yards.
Patriots quarterback Tom Brady was 21-of-36 passing for 237 yards, with two interceptions.
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