Keane slalomed his way through Dallas' defence in the 24th minute before unleashing a drive into the top left corner of the net to put Galaxy 1-0 ahead, and Gyasi Zardes doubled the hosts' advantage just before half-time.
Although the visitors hit back in the final half hour through Tesho Akindele, Galaxy held on for their first victory in five games, while Dallas' winless run stretched to six matches.
LA moved on to 12 points for the season to sit sixth in the MLS' Western Conference, just outside the play-off positions, while Dallas remain third on 17.
With key man Landon Donovan having joined the United States' squad for the FIFA World Cup, Galaxy lacked creativity early and barely fashioned a genuine chance until Keane's flash of inspiration.
In the 44th minute, Keane started a move that led to Galaxy's second goal, getting the ball on the left before teeing up Juninho, whose cross was headed home by Zardes.
In Wednesday's other MLS fixture, goals from Chris Rolfe and Fabian Espindola were enough for DC United to triumph 2-0 over Houston Dynamo.
