The hosts started the better of both sides and only had to wait 10 minutes to see their pressure pay off, as Feilhaber bent a free kick beyond goalkeeper Jaime Penedo.
It would have been worse for Galaxy later in the half if not for the intervention of Penedo, who had to be alert to deny Soony Saad's bid to add a second.
Kansas pounced on the missed chances and made Galaxy pay dear as Olum headed past Penedo just after the hour mark to double his side's advantage.
Keane then came within the width of the woodwork of halving the deficit by crashing against the crossbar, before the Irishman finally struck gold minutes later by converting a Bradford Jamieson pass beyond Andy Gruenebaum.
The former Tottenham Hotspur hitman had given his side some 10 minutes in which to force an equaliser and salvage an unlikely point, but City held out to boost their lead at the summit.