Austin Ekeler picked up a pair of one-yard rushing touchdowns for the visitors in the second and fourth quarters, while home quarterback Nick Foles had a horror debut for the Colts with three interceptions.
Justin Herbert started slowly but was solid for 235 yards through the air for the Chargers and importantly got the ball to Ekeler.
The Colts took advantage of two personal foul calls that led to the ejection of Pro Bowl safety Derwin James in the second quarter to get on the scoreboard for the only time courtesy of a 46-yard field goal by Chase McLaughlin.
Cameron Dicker potted a pair of field goals for Los Angeles in the second and third quarters, with Ekeler's second score ensuring the win halfway through the final quarter.
Questions are likely to be raised about the future of Jeff Saturday, who has been responsible for five consecutive losses since becoming interim coach of the Colts.
The Chargers, meanwhile, are riding high after three straight wins and securing a playoff spot ahead of Sunday's Battle of Los Angeles against reigning Super Bowl champions the Rams.