Sporting SM struck first in the 37th minute through midfielder Rodrigo Tello, who headed home a free header at the far post off a cross from Angel Valencia.
The Galaxy struck back, however, finding a 68th-minute leveler from Paintsil. The Ghanaian winger converted a comfortable finish after pouncing on a deflected ball in the penalty area.
LA narrowly avoided another concession 10 minutes from full time, when a long cross into the penalty box evaded goalkeeper JT Marcinkowski. But the effort rang off the crossbar, ensuring the Galaxy will head into Leg 2 next Wednesday at Dignity Health Sports Park on level terms with an all-important road goal.
The aggregate winner of the series advances to take on Jamaican Premier League side Mount Pleasant FA in the Round of 16.
