The Galaxy will meet the winner of the Eastern Conference final on Saturday between Chicago Fire and Real Salt Lake City in next Sunday's final in Seattle to determine the MLS champions.
The floodlights went out in the 18th minute, causing an almost 20-minute break, and the same happened again six minutes after the interval with stadium officials blaming the local grid for the failure.
Houston came close to taking the lead in the 78th minute when Brian Ching's header was spectacularly tipped over by L.A. keeper Donovan Ricketts and from the resulting corner Andrew Hainault headed home but the effort was ruled out for an infringement.
After the game ended goalless in regulation time, it was L.A. who went in front in the 13th minute of extra time when Beckham floated a free-kick into the box, Omar Gonzalez headed the ball down and defender Gregg Berhalter slid the ball home.
The Galaxy wrapped up the win when Houston midfielder Ricardo Clark brought down Alan Gordon and Landon Donovan confidently converted the penalty in the second period of extra time.
The Galaxy, MLS winners in 2005, are now one win away from being crowned champions in a season that saw Beckham jeered by his own fans amid reports of dressing room unease.
Beckham, who will join AC Milan on loan in January, said that period had helped the team gel.
"There is nothing wrong with a bit of controversy in a club - it brings players together and it did that," he told ESPN television.
"Its so important for us to do well, we have had so many disappointments over the past four years ... we have a great bunch of guys who work hard for each other," he added.
