Dallas (8-12-9) preserved its playoff hopes alive, improving to 3-1-1 in its last five to move four points behind Vancouver for the Western Conference's fifth and final postseason spot.
Mauro Rosales scored in the 20th minute for Seattle (12-6-8), which improved to 5-1-2 in its last eight MLS contests and edged past Los Angeles into third place in the West.
Hartman, the MLS' all-time leader in virtually every goalkeeping category, made three saves, including one on Sounders forward Fredy Montero, who was scoreless on five shots after getting six goals in his previous four league matches.
Rosales, who leads the MLS with 13 assists, put Seattle ahead with his second goal of the season after taking a beautiful setup from Steve Zakuani. Zakuani, making his first start and third appearance after recovering from a broken leg on April 22, 2011, ran around Dallas defender Zach Loyd on the left side before delivering a perfect feed across the goalmouth to the charging Rosales for an easy tap-in at the right post.
Jackson's third goal of the year tied it 12 minutes later. Blas Perez, making his first start and second appearance for Dallas after missing 12 games with a right foot sprain, delivered a nifty pass across the penalty area to the charging Jackson, who re-directed it home near the left post.
The Sounders nearly regained the lead just three minutes later, but Hartman made an outstanding leaping hand save on Brad Evans' point-blank header, barely deflecting the ball over the crossbar.
Seattle controlled the majority of the action through the first 20-plus minutes of the second half, but Jackson who had the best opportunity to snap the 1-1 tie, in the 69th minute. A short pass from Perez left Jackson alone with the ball about 10 yards out, but his point-blank shot was stopped by Sounders goalkeeper Michael Gspurning, who made a spectacular diving save.
Star forward Eddie Johnson, who is tied with Montero for the team lead with 11 goals, did not make the trip for the Sounders, remaining back in Seattle to rest.
Meanwhile, Dallas was missing speedy forward Fabian Castillo, who had scored goals in three straight games before leaving their last contest with a strained right hamstring.