After a scoreless first half, Joao Plata put RSL into the lead after latching on to Ned Grabavoy's deft touch and striking the ball underneath Galaxy goalkeeper Jaime Penedo.
But Keane levelled three minutes later, displaying physicality and silky skills to finish from just outside the area.
It was an imperious display from the Republic of Ireland international, going close many times before eventually putting Galaxy ahead in the 67th minute after latching onto Sean Franklin's cross
Defender Omar Gonzalez celebrated signing a new contract by scoring the side's third from a corner on 75 minutes before Keane completed his treble by chipping the onrushing 'keeper in the 86th minute.
Robbie Findley scored a consolation goal for RSL in the 95th minute but it was too little, too late.
The New York Red Bulls and the Philadelphia Union fought out a tight goalless draw at the Red Bull Arena.
A win for New York would have seen them leapfrog Sporting Kansas City at the top of Eastern Conference, who dropped points as Chris Wondolowski scored the only goal in the 55th minute as the San Jose Earthquakes defeated the leaders 1-0.
Italian veteran Marco Di Vaio took his season tally to 13 goals after his brace propelled the Montreal Impact to a 2-1 victory against DC United.
Di Vaio opened the scoring before Conor Doyle secured DC's equaliser on 81 minutes.
But the former Juventus star had the last say, firing in the winner only two minutes later.
Former English prodigy Giles Barnes scored twice as the Houston Dynamo ran out 3-1 winners over the Seattle Sounders.
In other results, Portland beat FC Dallas 2-1 and the Colorado Rapids were 2-0 winners over the Vancouver Whitecaps.
The New England Revolution defeated Chicago Fire by a 2-0 scoreline and Toronto had a sour road trip to Columbus, also losing 2-0.
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