Three goals in the opening 23 minutes, including a brace from Jesus Ferreira, set the Toros up for a critical three points after slip-ups from both Austin FC and Los Angeles FC.
Douglas Costa's well-taken free-kick in the 67th minute proved to be mere consolation, as the Galaxy failed to break Dallas down and force their way back into the contest.
Dallas now sit second in the West, a point off LAFC, who conceded two penalties in the opening 30 minutes to lose 2-0 to Colorado Rapids.
With the win, the Toros were able to go past Galaxy and Austin, who let slip a 1-0 lead at half-time following Daniel Pereira's red card to lose 2-1 to Real Salt Lake.
There were goals galore elsewhere in the West, with Portland Timbers giving Sporting KC a 7-2 trouncing while Houston Dynamo defeated Nashville SC 2-0. The Vancouver Whitecaps and SJ Earthquakes played out a 3-3 draw.
Elsewhere, Philadelphia Union's 1-1 draw with New York RB has made for up a three-way tie atop the Eastern Conference.
Daniel Gazdag put the Union ahead after some neat combination with Sergio Santos at the top of the penalty area in the 47th minute, but Luquinhas managed to restore parity with his goal in the 66th minute.
They are joined at the top of the East following respective wins for CF Montreal and Orlando City over Charlotte and Toronto FC.
New York City were able to make up ground with their 2-0 win over Columbus Crew and Cincinnati beat Chicago Fire 2-1, while Inter Miami and DC United played out a 2-2 draw.
After failing to score in their opening two games – a 1-0 loss to LA Galaxy and a goalless draw with Vancouver Whitecaps – City hit the lead after only seven minutes, with Alexander Callens rifling home after Montreal failed to clear their lines from a corner.
Santiago Rodriguez spectacularly chipped Sebastian Breza to double the margin on 20, but only after Montreal gifted City possession in an attempt to play out from their own half.
Zachary Brault-Guillard pulled one back for CFM, but following further City goals from Telles Magno and Thiago Andrade, the scoreline arguably belied the performance.
Douglas Costa opened his account in the MLS later on Saturday, scoring the equaliser in LA Galaxy's 3-2 loss away to Seattle Sounders.
Javier Hernandez gave the Galaxy a shock lead in the sixth minute, after fine link-up play between Raheem Edwards and Kevin Cabral, but the Sounders fought back to lead before the interval – with goals from Jordan Morris and Fredy Montero.
Costa drew the visitors level from the dead ball after 48 minutes, wrong-footing Stefan Frei, but Xavier Arreaga headed home in the 72nd to give the Sounders their first win of the season.
Los Angeles FC continued their unbeaten start to the season, meanwhile, grinding out a 2-0 win against a 10-man Inter Miami at DRV PNK Stadium.
A deflected Carlos Vela pass put Kwadwo Opoku through on the stroke of the interval, and the 20-year-old Ghanaian finished with composure. That didn't spell the end of events in the first half, though, as Brek Shea was given a straight red card in injury time.
LAFC had six shots for the match compared Inter's seven, despite 53 per cent possession and an extra man. However, an Ismael Tajouri free-kick crept inside Inters back post in the 82nd minute to secure the win.
Steve Cherundolo's side are equal top in the Western Conference with Real Salt Lake, who defeated New England Revolution 3-2. Austin FC couldn't keep pace in the final Saturday game, however, going down 1-0 away to Portland Timbers through a Bill Tuiloma goal.
Columbus Crew share the lead in the Eastern Conference, following their 2-1 win over Toronto FC. They are joined by Philadelphia Union, who defeated San Jose Earthquakes 2-0.
A Darwin Quintero brace handed Houston Dynamo a 2-0 win against Vancouver Whitecaps, while Brandon Vazquez scored a double of his own in FC Cincinnati's 2-1 win at Orlando City.
FC Dallas, Chicago Fire and Colorado Rapids respectively secured 2-0 victories over Nashville, DC United and Sporting Kansas City in Saturday's other games.
NYRB had to hold in final stages after leading by three goals at one stage, with goals from Sebastian Driussi and Ethan Finlay bringing Austin back in the contest, but Gerhard Struber's side deservedly took the win to keep up with Philadelphia Union and New York City.
First-half goals from Serge Ngoma and Dru Yearwood had the visitors up 2-1 at the interval after Driussi's first of the night, but Yearwood then set up Cameron Harper, who deftly cut inside Ruben Gabrielsen and finished to make it 3-1 after 51 minutes.
Substitute Tom Barlow made it 4-1 14 minutes later with a thumping finish on the half-volley straight from Carlos Coronel's goal-kick.
Following Union and City's wins on Saturday, NYRB restored their gap to the Eastern Conference's top two to five points with the win and joined Austin in earning the most wins on the road this season, with seven.
Los Angeles Galaxy marginally moved back into the Western Conference's playoff spots in Sunday's later game, defeating Atlanta United 2-0.
Needing the three points after losing their previous three games, the Galaxy hit the lead in only the seventh minute through Kevin Cabral.
They sought to absorb pressure in the second half with only 35.7 per cent possession, before substitute Dejan Joveljic eventually scored the sealer on the break in added time.
Greg Vanney's side moved into seventh in the Western Conference with the win, leapfrogging Portland Timbers despitte both being on 30 points, having secured two more wins.
