Dallas looked to be in complete control at half time after scoring three touchdowns in the second quarter, with Tony Romo combining with Dez Bryant twice before Sean Lee's 52-yard interception return made the score 21-10.
Nick Novak struck a 42-yard field goal in the final seconds of the first half, which helped prove the catalyst for a comeback from the hosts.
Rivers came into his own after the interval, quickly picking out Danny Woodhead in the end zone with a 13-yard touchdown pass.
That score edged the Chargers to within a single point, before Novak's trusty book edged San Diego ahead with a field goal.
Dallas struggled to make possession count in a tight fourth quarter and were punished ruthlessly by Rivers, who found Antonio Gates from 56 yards to secure a fine comeback win for the Chargers.