The Sounders are tied for second place in the Western Conference with Portland at 50 points apiece with one game left, two points behind West-leading Vancouver.
Dallas still has not clinched a playoff spot and is tied with the San Jose Earthquakes for the final position at 43 points after Sunday's games.
With Seattle clinging to the lead on Victor Rodriguez's 31st-minute goal, Bruin made it 2-0 in the 64th minute. The initial shot by Clint Dempsey was saved by goalkeeper Jesse Gonzalez, but Gonzalez was not able to gain control of the rebound. Bruin, with defender Atiba Harris hanging onto him, still got to the ball at the top of the six-yard box and punched it into the back-left corner.
Late in the 67th minute, Bruin was playing just to the left of the penalty spot when the ball was deflected his way, and he fired it into the back left side. He now has 10 goals for the season.
Lamar Neagle added a stoppage-time goal for Seattle.
Sebastian Blanco scored a pair of second-half goals to lead the Portland Timbers over D.C. United 4-0 and into the playoffs.
Near the end of the first half, Portland forward Darren Mattocks intercepted a weak backpass by defender Chris Korb, then was taken down by goalkeeper Steve Clark for a penalty. Diego Valeri knocked his shot past Clark and into the lower left corner of the net for his 21st goal. Valeri has scored in 10 of the last 11 games.
Portland goalkeeper Jeff Attinella preserved the lead in the 49th minute with two saves in quick succession, first diving to keep out a header by Luciano Acosta, then blocking a rebound shot by Patrick Mullins.
A minute later, Portland doubled the lead. Alvas Powell dribbled in from the right wing and, with no defenders attempting to close him down, unleashed a left-footed shot past a diving Clark for his second of the season.
The Timbers extended the lead to 3-0 in the 60th minute. Valeri showed off his juggling skills while controlling a long pass, then exchanged passes with Blanco on the edge of the penalty area, creating enough space for Blanco to rip his own left-footed shot inside the far post.
Blanco capped the scoring in the 86th minute, picking up a loose ball and dribbling past multiple defenders and Clark before knocking in his eight of the season.
Nemanja Nikolic scored two of his three goals in the second half, increasing his league-leading total to 24, as the Chicago Fire rallied for a 3-2 win over the Philadelphia Union.
The Fire moved into third in the Eastern Conference, which would be good for a home game in the playoffs. Nikolic is four goals clear of his pursuers for the Golden Boot with one game to play.
Nikolic opened the scoring in the third minute, getting to a long feed from Brandon Vincent and aided by a bad angle by defender Jack Elliot before beating Andre Blake.
Nikolic's penalty kick in the 64th minute tied it at two-all after David Accam was taken down in the penalty area. Nikolic completed his first career hat trick with an easy finish from the middle of the box on a feed from Brandon Vincent for the game winner in the 78th.
Chicago defender Johnan Kappelhof's own goal put the Union in a 1-1 tie at the six-minute mark and a feed from Haris Medunjanin to Alejandro Bedoya for a chip shot was good for the lead from the 13th minute.
Philadelphia finished 1-10-6 on the road this season.
Diego Fagundez scored two goals in the second half as the New England Revolution beat 10-man New York City FC 2-1.
New England set a club record with 12 wins at home. NYCFC had several possibilities to clinch a bye in the playoffs but needed at least a draw.
Fagundez opened the scoring in the 51st minute and made it 2-0 in the second minute of stoppage time with his sixth goal of the season. Andrew Farrell set up the first goal by dribbling into the area and finding Fagundez at the top of the box. It was his first goal since June 3. Fagundez's next goal was a tap-in after his breakaway shot was knocked back to him.
Khiry Shelton scored for NYCFC in the third minute of extra time by getting past the defense for a long ball and heading it in.
NYC's Jack Harrison was shown a straight red card in the 26th minute for a late challenge on Teal Bunbury. It was Harrison's first red card of his professional career and he'll miss the regular-season finale against Columbus Crew SC next weekend.
Ola Kamara scored his 17th goal of the season on his birthday as Columbus Crew SC beat Orlando City 1-0 to spoil Kaka's final game with the Lions.
Kaka, the 35-year-old Brazilian star, announced on Wednesday he would not re-sign with Orlando City after a three-year run with the expansion team.
"A lot of emotions today," Kaka said after the game. "It was a special day for me because the whole day keeps reminding the past three years -- everything that already happened in this city, with this club."
Kamara scored on an easy redirection in the 67th minute. Federico Higuain found an open Hector Jimenez along the left side for a cross in front of goal to an unmarked Kamara.
Columbus sits in fifth in the Eastern Conference, three points back of second-place New York City FC.
Zack Steffen kept Orlando City scoreless for his ninth shutout, which is the most by a goalkeeper in his first year of MLS play.
Brad Guzan and Luis Robles each made three saves as Atlanta United and the New York Red Bulls -- both bound for the playoffs -- battled to a 0-0 draw.
Atlanta could have moved into a tie for second in the Eastern Conference with a chance to earn a first-round bye when the regular season ends next Sunday. A win would have guaranteed United a home game in the knockout round.
Guzan picked up his eighth shutout for expansion Atlanta with a couple of big saves in the wide-open game, including robbing Bradley Wright-Phillips from point-blank range in the 65th minute. Guzan also had a diving save in the 73rd minute to deny Gonzalo Veron with Josef Martinez clearing the rebound.
Robles earned his 10th clean sheet for the Red Bulls. His key save was in the sixth minute on a diving stop of Yamil Asad's close-range header.
The teams had a combined 22 shots, 13 by the Red Bulls.
Neagle's header puts Seattle ahead by four
Major League Soccer: Lamar Neagle (90'+2') Seattle Sounders 4-0 FC Dallas
Chicago 3-2 Philadelphia: Nikolic the hat-trick hero
Nemanja Nikolic netted his league-leading 24th goal of the season to lift Chicago to a 3-2 win over Philadelphia.
New England 2-1 NYCFC: Fagundez at the double
Diego Fagundez scored a brace for New England as they got their 12th win at home, while NYCFC failed to secure a playoff bye.
Orlando 0-1 Columbus: Lions lose Kaka farewell
Birthday boy Ola Kamara spoiled Kaka's final match at Orlando City Stadium, picking up three points for Columbus.
NY Red Bulls 0-0 Atlanta: Big saves in scoreless draw
Luis Robles and Brad Guzan both made big stops as New York and Atlanta played to a scoreless draw.
Portland 4-0 D.C. United: Timbers cruise to playoffs
Portland Timbers made easy work of D.C. United to book a playoff spot after missing out in 2016.