Brett Gardner drew a leadoff walk and beat the throw to second on Kevin Reese's sacrifice bunt before Basak drilled the winning double to left.
Basak also scored in the fourth on a passed ball after reaching on a bunt single.
T.J. Beam (3-2) allowed two hits and struck out two in two scoreless innings for the win. Starter Ben Kozlowski allowed a run on three hits and three walks while fanning three over three frames for the Yankees (66-50), who won the opener, 4-0.
Jed Lowrie drove in both runs for the Red Sox (53-63) with a double in the third and a single in the fifth. Jacoby Ellsbury went 3-for-4 with his International League-leading 29th stolen base and two runs scored.
Pawtucket reliever Javier Lopez (1-1) retired one batter and surrendered two runs on one hit and a walk. Starter John Barnes allowed one unearned run on three hits and seven walks with six strikeouts over six innings.