The home side exerted early pressure on the Whitecaps' goal, but Stefan Ishizaki was unable to direct his effort from outside the box on target.
Vancouver hit back with an opening of their own as Manneh went on a surging run down the left and his strike went just past the post.
The Whitecaps took the lead in the 32nd minute when Manneh speeded down the left and raided into the box, and this time his right-footed strike beat Brian Rowe and nestled in the bottom corner.
Cristian Techera almost doubled his side's advantage when he latched onto a pass from Manneh, but his strike clattered off the post and bounced to safety.
In the second half, Manneh caused problems for the home side again as he weaved his way through two defenders into the box, but Rowe made a crucial block to deny the forward.
Robbie Keane almost brought the Galaxy level as he controlled a looping cross into the penalty area, only for his shot to go wide of the target.
The visitors came within inches of doubling their lead as they hit the Galaxy on the counter, but Pedro Morales's powerful strike hit the post and was cleared by the home defence.
The hosts pushed forward for an equaliser in the dying minutes, but the Whitecaps held out for the three points to move level on points with the Seattle Sounders at the top of the Western Conference.