Burnett (11-11) surrendered a pair of runs on five hits, walking two and fanning six for the American League East champion Yankees.
Jorge Posada smacked a two-run homer as the Yankees added to the woes of the AL wild card-leading Red Sox, who stumbled to their 18th loss in 23 games.
The Game 1 loss, coupled with Tampa Bay's victory over Toronto Sunday, reduced Boston's wild card lead to a mere half-game. Another setback in Sunday's nightcap would eliminate the Red Sox's once-sizable advantage in the wild card chase.
Boston's Tim Wakefield (7-8) lasted four innings, yielding five runs on five hits.
Jacoby Ellsbury socked a pair of solo homers in the loss.