Rivers passed for 434 yards and three second-half touchdowns, including a 27-yarder to Tyrell Williams that gave Los Angeles a 16-0 lead late in the third quarter.
Dallas broke a 10-quarter drought without a touchdown early in the fourth when running back Rod Smith powered two yards for a score. Smith's touchdown capped a nine-play, 81-yard march.
But Cowboys quarterback Dak Prescott threw incomplete on the two-point conversion attempt and the Chargers maintained a 16-6 lead.
Dallas' offensive woes continued midway through the fourth quarter when Los Angeles cornerback Desmond King returned an interception 90 yards for a touchdown.
Wide receiver Keenan Allen burned the Dallas defense as he caught 11 passes for 172 yards and a touchdown to lead the Chargers (5-6).
Dallas (5-6) dropped its second home game of the week after Philadelphia handed the Cowboys a 37-9 defeat on Sunday night. The Cowboys are 2-4 at home this season.
Los Angeles established a two-score lead when Rivers found tight end Hunter Henry for a 3-yard touchdown. Rivers looked off his first option and turned to Henry, then threw to Henry's right, away from Cowboys safety Byron Jones.
Chargers kicker Nick Novak missed the extra point, holding Los Angeles' advantage to 9-0 with 8:59 left in the third quarter.
Anyone napping during the first half didn't miss much offense as neither team scored through the game's first 28 minutes.
Los Angeles reached the Cowboys' 17 on its first possession, but Novak missed a 35-yard field goal attempt.
Novak appeared to be injured on the Chargers sideline, prompting punter Drew Kaser to warm up by kicking into a net, which he missed on one attempt.
Kaser successfully kicked an extra point following the Chargers' second touchdown. He went 1-for-3 on extra-point attempts.
Los Angeles chose to go for a fourth-and-13 from the Dallas 16 early in the second quarter. Rivers completed a pass to running back Austin Ekeler, but Cowboys cornerback Jourdan Lewis stopped Ekeler after a 12-yard gain.
Novak came back into the contest and booted a 22-yard field goal for the game's initial score with 1:56 remaining in the second quarter.
Dallas failed to travel more than 28 yards on any of its first-half possessions, resulting in four punts. The Cowboys managed just 84 yards total offense in the first half.
NOTES: Dallas lost its last two games and managed just one touchdown with LT Tyron Smith out of the lineup. In that time, Cowboys QB Dak Prescott was sacked 12 times. Smith started on Thursday. ... Chargers first-year head coach Anthony Lynn is a native Texan and played high school football at small-town powerhouse Celina, about 50 miles north of AT&T Stadium. ... The Cowboys and Chargers met for the 11th time and the first time on Thanksgiving Day. Dallas has a 6-5 edge in the series, though the Chargers have now won four of the last five. ... Despite losing RB Ezekiel Elliott to a six-game suspension two weeks ago for violating of the NFL's personal-conduct policy, Cowboys RB Darren McFadden was a healthy scratch on Thursday for the second-straight week. Dallas went into Thursday's game with two running backs (Alfred Morris and Rod Smith) and a fullback (Keith Smith) on its active roster.