Juninho tucked away a feed in the 66th minute from Jose Villarreal after the rookie forward stripped Dallas defender Matt Hedges about 15 yards outside the Hoops' box and raced past goalkeeper Kevin Hartman on the left. Juninho first-timed the shot into a virtually open net.
Dunivant tallied in the 84th after sending a pass to Robbie Keane in Dallas' box, then racing forward for a penetrating ball from Michael Stephens.
The victory moved the Galaxy (12-11-4) within three points of second-place Seattle and third-place Real Salt Lake in a tight battle behind Western Conference leader San Jose. Dallas (8-12-8) is five points behind Vancouver in the fight for the fifth and final Western playoff berth.
''I think (our defense) was real good,'' said Galaxy coach Bruce Arena, whose team has six shutouts in its last 14 league games. ''A real solid effort. This is a real good team we beat today. They were certainly ready to play. They pushed us hard. These are the kind of games we have to be able to win: capitalize on a mistake, get a goal, and do your defending.''
Juninho has five goals in the past four games, including a pair Thursday night in a CONCACAF Champions League victory over El Salvador's Isidro Metapan.
The match was a defensive battle marked by solid play from both backlines until Hedges' error. The Galaxy struggled to turn its advantage in possession into legitimate chances, with Hedges and fellow center back George John repeatedly cutting off passing lanes and disrupting attackers, and Dallas' speed up front was stymied by LA's backline, led by Omar Gonzalez and A.J. DeLaGarza.
''L.A. does a really good job of recovering,'' Dallas coach Schellas Hyndman said. ''We know that when they get a lead, they're really hard to break (down).''
The Galaxy were without captain Landon Donovan, who is nursing a hamstring strain, and David Beckham lasted until the 78th minute despite a stomach virus that had him awake and vomiting most of Saturday night.
Dallas' tactical plan suffered after Fabian Castillo exited near the end of the first half. He strained a hamstring leaping to avoid Galaxy goalkeeper Josh Saunders following a through ball from David Ferreira.
The Hoops' other chance was in the 61st minute, when Ferreira beat several defenders into the Galaxy box, but Saunders poked the ball away.