Teague capped off a big night in which he also made a 3-pointer with five seconds to play in the first overtime that brought about a 108-108 tie.
The teams combined to score 36 points in the second five-minute extra period and Teague finished things off after Atlanta's Pero Antic won a jump ball for the Hawks with 25 seconds to go.
Four Atlanta players reached the 20-point mark with Teague producing a career-best 34 to go with 14 assists. Al Horford added 25 and both Kyle Korver and Paul Millsap had 20. Horford, however, left the game late in the second half with an injured right shoulder.
The Hawks overcame a 40-point show by Kyrie Irving, who attempted 33 shots and handed out nine assists.
Cleveland lost its third straight.