The Giants trailed by 29 points at half time and looked to be no match for the Bulldogs' powerful midfield, but a knee injury to Dogs defender Josh Bruce early in the third term changed the momentum entirely, with the Giants running out winners 11.12 (78) to 10.13 (73) at Mars Stadium.
Bruce, who missed the bulk of the 2022 season after injuring his ACL in 2021, fell to the ground clutching his knee after turning awkwardly, compounding an earlier injury to defender Sam Keath who was subbed out in the opening half with concussion.
The Giants had managed just two goals in the opening half but Greene took full advantage of the shift in momentum, exploding to kick four goals and set up another in an electric third term.
The intensity from both teams rose in a tight and tough fourth quarter, but the Giants continued to push hard, with goals to Jake Riccardi and a fifth to Greene putting GWS in front with five minutes to go.
A 40-metre drop punt from Brent Daniels from close to the boundary stretched the margin out to 11 points, but Bailey Dale hit back with a goal for the Dogs moments later.
The Bulldogs had a number of chances to snatch victory in the final moments, but the Giants' defenders kept them at bay and helped their side to the first seven-win streak in club history.
Star back Sam Taylor (15 intercept possessions) played a crucial role in the win, helping to halt the influence of the Bulldogs' multi-pronged forward line after an opening quarter in which the Bulldogs took nine marks inside 50.
Veteran Callan Ward also played a key role, slowing Marcus Bontempelli down after a commanding first quarter in which he notched up 13 disposals and set up the Dogs' first three shots at goal.
Lachie Whitfield (34 disposals), Stephen Coniglio (34), and Lachie Ash (30) were key contributors, while Jake Riccardi kicked three goals.
A finals berth appeared extremely unlikely when the Giants had just four wins on the board heading into Round 13, but with seven consecutive wins they now hold the AFL's biggest current winning streak.
The victory in Ballarat also added to the Giants' impressive away record, having won five consecutive matches outside Sydney.
The Giants will be hoping to extend their winning streak to eight matches when they meet Sydney in the Battle of the Bridge at GIANTS Stadium next Saturday night, before facing Port Adelaide, Essendon and Carlton to round out their home and away season.
The Bulldogs will be hoping to bounce back against Richmond at Marvel Stadium on Friday night, with games against Hawthorn, West Coast and Geelong still to come.
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