The Flames have won three in a row overall and have won eight straight at home and are 10-0-1 in their past 11 at the Saddledome. Their last home regulation loss was Nov. 15 against Ottawa.
Anaheim had its three-game winning streak snapped.
Kiprusoff improved to 20-14-2 on the season and earned his third shutout of the season and 43rd of his career, tying him for 27th on the career list with Lorne (Gump) Worsley.
Jones converted a rebound of his own shot and flipped the puck over Jonas Hiller for the win.
Jones was playing in his third game with the Flames since he was acquired from the Tampa Bay Lightning on Jan. 6.
Earlier on Thursday, Calgary reacquired forward Mike Cammalleri, who led the Flames with 39 goals in 2008-09, in a trade with Montreal.
The Flames' lineup could have used his offensive prowess on Thursday.
They had six shots through the first 40 minutes. However, Anaheim couldn't solve Kiprusoff either despite holding an edge in shots.
Forward Rene Bourque, who was serving the fourth game of a five-game suspension, was the key player sent to Montreal in the Cammalleri deal. Bourque was suspended for elbowing Washington's Nicklas Backstrom in the head on Jan. 3.
Flames defenseman Mark Giordano assisted on the winner in his first game back after missing 21 with a hamstring injury. The assistant captain had not played since Nov. 29.
Anaheim has dominated Calgary for the past several years, going 20-1-4 and picking up points in 24 of the 25 games between the clubs dating back to March 1999.
A video review was required on two Anaheim scoring chances.
A review confirmed the ruling.
Kiprusoff jabbed out his pad to stop it, but momentum carried Kiprusoff into the net and there was a question whether the puck crossed the line.
A review showed the puck stayed out.
Hiller got the surprise start for the Ducks. Earlier in the day, Ducks coach Bruce Boudreau said Jeff Deslauriers was going to play against the Flames with Hiller to start against Edmonton on Friday night.
Hiller was not busy for the first 40 minutes, facing just six shots. He was busier in the third period and overtime as he faced 13 shots.
One of the chances in the third was a hard slap shot from Flames defenseman Anton Babchuk, which deflected off his helmet and went into the crowd.
Notes: It was Calgary's first game against a Pacific Division opponent. Four of the Flames next five games are also against the Pacific Division... The Flames' record for most consecutive wins at home is 10, which was set Nov. 7 to Dec. 12, 2006... Calgary D Chris Butler played in career game No. 200. ... It was career win No. 296 for Kiprusoff, which ties him with Ron Hextall for 27th on the career list. ... Anaheim is 1-16-3 when giving up the first goal. ... Boudreau is 7-9-3 since taking over as Ducks coach on Nov. 30.