Tyler Arnason set up a frenetic finish when he scored with 2:01 left, and the Avalanche had a power play for the final 1:17 after Mattias Norstrom was called for high sticking, but Colorado couldn't tie it.
Avs goalie Jose Theodore, who was seeking his first four-game winning streak since Oct. 31 to Nov. 8, 2005, with Montreal, allowed four goals in the second period and was replaced by Peter Budaj with Colorado trailing 5-2.
Avery's second goal came in the third period and proved to be the game-winner.
After Conroy's goal put the Kings ahead 1-0, the Avs outshot the Kings 7-0 over the final 4:42 of the first period, and two of those went past Dan Cloutier to give Colorado a 2-1 advantage. Marek Svatos tied it 1-1 with 3:52 left when he scored on a power play with an assist from Joe Sakic, and 24 seconds later Wojtek Wolski scored off a give-and-go with Ossi Vaananen.
The momentum quickly shifted, however, when Lubomir Visnovsky tied it 2-2 just 2:20 into the second period, and then the Kings scored two goals a minute apart to take control.
Avery gave the Kings the lead with a short-handed goal off an assist from Alexander Frolov, his first of three assists. A minute later, Thornton received the puck from Michael Cammalleri and put it past Theodore for a 4-2 lead.
Frolov assisted on Anze Kopitar's goal with two minutes left in the period that made it 5-2.
Brett McLean's goal 90 seconds into the third cut Colorado's deficit to 5-3, and Sakic scored at 12:08 to pull the Avalanche to 5-4. Sakic tied Ray Bourque (1,506 with Boston) for the fifth-most points for a single franchise in NHL history.
Just more than a minute later, however, Avery's second goal made it 6-4.
Notes: Brown had a roughing penalty that sent John-Michael Liles to the dressing room in the second period. ... Sakic needs one assist to tie Stan Mikita (926) for 15th place on the NHL's career list. ... Theodore saved 17 of 22 shots. In his previous three games, he stopped 97 of 103.