Fitzpatrick amassed just 86 yards passing but connected on a 9-yard TD toss to Terrell Owens that opened the scoring and sent the Bills (5-8) ahead to stay late in the first quarter.
Lindell kicked field goals of 41, 24 and 36 yards, Fred Jackson ran for 99 yards and Marshawn Lynch ran for 84 for Buffalo.
Bills defenders intercepted four passes by Kansas City's Matt Cassel.
The Chiefs (3-10) scored on a 76-yard run by Jamaal Charles and a 21-yard field goal by Ryan Succop.
Cassel was 26-of-43 passing for 224 yards. One of his intercepted passes was picked off in the end zone.
Charles rushed for 143 yards.
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