In a meeting of struggling teams, the Chargers produced 21 points and two defensive scores in the final period to end a three-game losing streak. The Chiefs' current skid was extended to five contests.
Kansas City fell to 1-7 and the Chargers climbed to 4-4, moving within a half game of AFC-leading Denver.
Both offenses were sluggish through three quarters, at the end of which San Diego held only a 10-6 lead.
Philip Rivers then drove the Chargers 80 yards in eight plays and his 13-yard scoring pass to Floyd with 11:33 remaining gave San Diego some breathing room.
On the second play after the kickoff, linebacker Jarret Johnson sacked Kansas City quarterback Matt Cassel in the end zone and Phillips recovered the resulting fumble for a touchdown.
San Diego's Demorrio Williams soon added to Cassel's woes by intercepting a tipped pass and returning it 59 yards for another score.
Cassel started in place of Brady Quinn, who is out of action with a head injury. Cassel completed 19-of-29 throws for 181 yards and directed the Chiefs to their only touchdown in the game's final minutes.
Rivers, who also threw a first-period touchdown of 14 yards to Antonio Gates, was 18-of-20 for 220 yards. One of his two misses was an interception thrown with his team on the Kansas City 1-yard line in the final seconds of the first half.
Beyonce flaunts bikini body, Blue Ivy in vacation pics
Jordana Brewster on Paul Walker: 'He was an enormous presence in my life'