Bingham was stellar prior to leaving the match due to an apparent hamstring injury, while LA's Jonathan Klinsmann turned aside one shot -- a point-blank attempt from Jaroslaw Niezgoda in the 89th minute -- to record his third career shutout.
Justin Rasmussen elevated a shot over the crossbar in the 91st minute for the short-handed Timbers (1-3-1, four points), who played without seven players due to injury and one due to international competition. Portland saw its winless stretch extend to four games (0-3-1) following its 1-0 season-opening victory over Sporting Kansas City on Feb. 27.
Klinsmann got the nod in place of Jonathan Bond, who sustained a shoulder injury in the Galaxy's 1-1 draw versus the Vancouver Whitecaps last Saturday. The Galaxy (0-1-3, three points) settled for their third straight draw overall and are 3-0-2 in their last five matches against the Timbers.
The Galaxy enjoyed the better of play, including a decisive 10-1 advantage in shots in the first half while nearly scoring on three point-blank opportunities.
Gaston Brugman's right-footed shot from just outside the box was tipped away by a leaping Bingham to force a corner kick in the sixth minute.
Preston Judd nearly made the most of his first MLS start when his left-footed attempt caromed off the right post in the 14th minute, and Riqui Puig's point-blank shot was tipped away near the far post in the 29th minute.
The Galaxy were snake-bitten again in the 57th minute, as Memo Rodriguez's direct free kick banked off the left post.
The physical match featured eight yellow cards, with five being issued to LA.