The Stars had been 6-3 in shootouts this season and 18-4 all time in the tiebreaker, but Norrena denied Sergei Zubov, Mike Modano and Jussi Jokinen to give the Blue Jackets their third shootout win in eight tries this season.
Chimera beat Dallas goalie Marty Turco for the only shootout conversion. Jokinen missed the net on his attempt to send the shootout into a fourth round, and the Blue Jackets went onto the ice to celebrate.
Modano earned his 497th career goal and Darryl Sydor had a goal and an assist for Dallas, which had been 12-3-3 in its previous 18 games including an 8-1-1 run at home. The Stars were playing in their fourth straight overtime game.
Modano's second-period goal, his first in eight games and 12th of the season to pull the Stars even after falling into a 2-0 deficit, leaves him five behind Joe Mullen for the all-time record among U.S.-born players.
Modano got his second goal in 13 games since missing 23 straight games because of a hip injury.
Manny Malhotra and Sergei Fedorov had goals, and Norrena stopped 22 shots for the Blue Jackets, 4-8-2 in their past 14 games.
The Stars were on the second leg of back-to-back games and were playing their third game in four nights. Dallas lost on the road in overtime to the Florida Panthers on Thursday night, 2-1.
Former Stars coach Ken Hitchcock won in Dallas behind the Columbus bench for the second time in as many trips this season. Columbus beat Dallas 3-1 on Dec. 12.
Malhotra, who played for the Stars from 2002-03, scored on a one-timer from the slot off a pass from Chimera at 11:34 of the first period to put the Blue Jackets in front. Turco had fallen and was out of position on Malhotra's seventh goal of the season.
Columbus took a 2-0 lead at 5:21 of the second period when Fedorov pounced on a rebound for his 17th goal of the season.
Sydor pulled Dallas within 2-1 at 8:03 of the second period with a power-play shot from the right point for his fifth goal of the season.
Modano tied it at two about a minute and a half after Sydor's goal when Modano converted Jokinen's centring pass.
Notes: Mattias Norstrom was among Dallas' scratches. Norstrom has yet to make his Stars debut after being acquired on Tuesday in a trade with the Los Angeles Kings. ... Nolan Baumgartner played in his first game for Dallas after being claimed on waivers from the Philadelphia Flyers on Saturday. ... Alexander Svitov was scratched for Columbus because of an upper-body injury. ... Adam Foote of the Blue Jackets left the ice 1:34 after the opening faceoff with a lower-body injury and did not return. Foote had been checked into the boards by Modano and was helped from the ice favouring his left leg. ... Stars defenceman Stephane Robidas set a career high with his 16th assist.