Salt Lake goalkeeper Nick Rimando was in inspired form to frustrate the Galaxy, who were eventually undone by Joao Plata's 80th minute goal.
The Seattle Sounders left it even later to score the winner in their 1-0 victory over Sporting Kansas City, with substitute Chad Barrett scoring in stoppage time.
Federico Higuain netted twice in a 3-0 win for the Columbus Crew over DC United.
Jairo Arrieta opened the scoring before Higuain got a second from the penalty spot in the 27th minute before sealing it with a stoppage-time strike.
The Houston Dynamo got their campaign under way with a convincing 4-0 victory over the New England Revolution.
Will Bruin scored twice in the opening 13 minutes before Boniek Garcia made it 3-0 with 23 minutes gone. Omar Cummings added the gloss late on.
There was also a big win for the Vancouver Whitecaps as they beat the New York Red Bulls 4-1.
Kenny Miller opened the scoring from the penalty spot after 34 minutes. Sebastian Fernandez then doubled the lead four minutes into the second half before Miller's second with 13 minutes left.
Pedro Morales added a fourth late on before Bradley Wright-Phillips got New York on the board at the death.
FC Dallas came from behind to beat the Montreal Impact 3-2.
The Canadian side led through Sanna Nyassi after 10 minutes, but Fabian Castillo cancelled that out six minutes later and Blas Perez had the Texans in front with a penalty in the 24th minute.
Mauro Diaz added a third at the start of the second half, and Andrew Wenger's 65th minute strike was not enough for Montreal.
Gaston Fernandez scored in the 90th minute to salvage a 1-1 draw for the Portland Timbers against the Philadelphia Union, who had led through Jack McInerney's 64th minute opener.