England international Sancho, 19, brought Dortmund to life in the second half, driving home a much-needed equaliser before grabbing an injury-time assist to seal Dortmund's comeback.
"We stayed patient, we wanted to play with a bit more tempo in the second half," Dortmund captain Marco Reus told broadcaster DAZN.
Cologne, celebrating the first home game since their return to the top flight, played with furious energy early on and were rewarded when Dominick Drexler headed in a corner at the far post on 29 minutes.
Dortmund, who got their title charge off to a flying start with a 5-1 win at home to Augsburg last week, looked lacklustre by contrast.
Frustrated by fine performances from Cologne's new defensive signings Kingsley Ehizibue and Sebastiaan Bornauw in the first half, the visitors huffed and puffed in the second until Sancho finally smashed down the door on 70 minutes.
Having skimmed the top of the net just seconds earlier, Sancho found himself in space as Dortmund took a quickfire corner, and fizzed a low shot inside the far post.
The goal was Sancho's 15th in the Bundesliga since joining Dortmund in 2017, making him the youngest player to reach that mark in the German league.
The Englishman was then involved at the start of a move which saw Achraf Hakimi head Dortmund in front from close range with four minutes to play.
With Cologne probing for an equaliser deep into injury time, Sancho burst away on the counter-attack and squared the ball for Paco Alcacer to tap in Dortmund's third.