Mirza Teletovic came up with a career-best 34 points in pushing the Nets to their fourth straight win and their ninth victory in 10 games.
Dallas trailed by three when Dirk Nowitzki was fouled with 7.1 seconds remaining. He made his first foul shot, but missed the second on purpose in hopes a teammate could claim the rebound.
Kirilenko grabbed the ball instead and his free throws put Brooklyn in front by four, giving the Nets a cushion they needed when Monta Ellis hit a 3-pointer at the buzzer.
Teletovic scored 24 of his points in the second quarter while lifting Brooklyn to an 11-point lead at the break.
Deron Williams had 18 points and 11 assists off the bench as part of the Nets' seventh straight home win.
Dallas was led by Vince Carter with 19 points and Nowitzki had 18.