Cole forged a 1-1 tie with a first-period goal and Tyler Seguin supplied a go-ahead score for Dallas with 7:47 remaining in the second.
Cole followed with his second of game 3 minutes later and the Stars held on to take a fifth win in seven games.
Kari Lehtonen stopped 23-of-25 shots en route to the triumph.
The Kings got goals from Jeff Carter and Anze Kopitar but couldn't avoid a fourth straight loss.
Ben Scrivens made 29 saves for Los Angeles.