The NFC East champions' defence frustrated Seattle's rushing attack for most of the game and ended their run of winning in their last nine opening play-off games.
Ezekiel Elliott rushed for 137 yards and the go-ahead fourth-quarter touchdown and finished his first playoff win with 30 touches for 169 yards.
Quarterback Dak Prescott also made a big impression, completing 22 of 33 passes for 226 yards and ended with a 16-yard scramble to set up a touchdown with 2.07 left on the clock.
Seattle refused to quit and Tyler Lockett quickly followed with a touchdown off a 53-yard catch but the Seahawks' fate was sealed when Cole Beasley caught Michael Dickson's drop kick.
Dallas will now face either the New Orleans Saints or the Los Angeles Rams in the divisional round.
The Colts ran riot over the AFC South champions, building a 21-0 lead by half-time, while playoff debutant Deshaun Watson was sacked three times and completed 29 of 49 passes for 235 yards and a touchdown.
Andrew Luck threw for 222 yards and two touchdowns, helping set up a divisional-round clash against top seed the Kansas City Chiefs.