Galaxy sweep Rapids with convincing away win to advance in MLS play-offs
Brazil's Gabriel Pec fired the visiting Galaxy in front in the eighth minute and after Oliver Larraz equalized in the 19th, Joseph Paintsil put Los Angeles 2-1 up in first-half added time.
Spain's Puig plundered two goals in the space of two minutes in second-half injury time to seal the victory.
The Galaxy's five MLS Cup titles are a record but their last came a decade ago.
They beat the Rapids 5-0 in the opening match of the best-of-three series and now await the winner of the series between Real Salt Lake and Minnesota United, which Minnesota lead 1-0.
In Charlotte, goalkeeper Kristijan Kahlina starred as the hosts beat Orlando City 3-1 on penalties to level their series after a match that ended in a goalless draw.
Orlando City had won the series opener 2-0, but after Charlotte's first ever playoff victory it will come down to game three in Orlando on November 9.
The winner advances to an Eastern Conference semi-final against either Atlanta or Lionel Messi-led regular-season champions Miami.
Kahlina made two saves in the penalty shootout - denying Nicolas Lodeiro and Duncan McGuire - and an effort from Orlando's Robin Jansson sailed high.
Charlotte found the net on their first three attempts from Patrick Agyemang, Karol Swiderski and Ashley Westwood for the victory.