Fredy Montero scored for the Western Conference-leading Sounders, who snapped a two-match losing streak.
Dallas extended its unbeaten streak to three matches, which has helped it climb out of the conference cellar.
The tying goal came in the first of three minutes of stoppage time. A Dallas cross was headed toward the right wing corner by Sounders defender Xavier Arreaga but the ball fell right to Nkosi Burgess, who hammered the ball toward the front of the net and Jara was able to make a sliding deflection from about two yards for the equalizer.
Dallas goalkeeper Jimmy Maurer made five saves, all in the first half, to keep his team in the game.
Montero opened the scoring in the 72nd minute, just before coming off for a substitute.
Montero's goal came off a free kick by Joao Paulo on the left wing. Paulo's cross found Montero just outside the top of the 6-yard box and Montero was able to go up over a defender and put a powerful header into the top left corner of the net, just out of the reach of a leaping Maurer.
Montero, in his second stint with the Sounders, extended his franchise record with his 51st career MLS goal with the club.
The Sounders, who set an MLS record with a 13-game unbeaten streak to start the season, got a boost from the return of midfielder Cristian Roldan, who helped the U.S. win the championship of the CONCACAF Gold Cup. All three of Seattle's losses came while Roldan was on international duty.
Sounders captain Nicolas Lodeiro, a member of MLS's Best XI last season, came on as a substitute late in the match. Lodeiro played just 24 minutes this season before undergoing arthroscopic knee surgery.