The old record was held by Oakland's Rich Gannon, who completed 21 straight against Denver on Nov. 11, 2002.
Brunell, 36, found Santana Moss on 6-yard pass late in the third quarter for his 22nd completion of the game.
He finished 24-of-27 for 261 yards and a touchdown in the Redskins' 31-15 win.
Donovan McNabb holds the NFL record for most consecutive passes competed over a two-game span with 24. That was 10 straight Nov. 28, 2004, against the Giants and 14 in a row Dec. 5, 2004, against Green Bay.