Dominick Drexler's header gave Cologne a first-half lead in its first home game since returning to the Bundesliga after last season's promotion.
But Sancho, who also scored in last weekend's 5-1 rout of Augsburg, hauled Lucien Favre's side level in the 70th minute.
The England winger collected a short corner from Thorgan Hazard then advanced towards the box and whipped a wonderful low finish across goalkeeper Timo Horn.
Substitute Achraf Hakimi put Dortmund ahead with four minutes to go, meeting Lukasz Piszczek's right-wing centre with a towering back-post header, then Sancho led a break to set up Paco Alcacer in added time.
Sancho's 15th Bundesliga goal means the former Manchester City talent has become the youngest player to reach that figure in the league's history.
He takes the record from Horst Koppel, who broke through at Stuttgart in the 1960s and went on to enjoy two spells with Borussia Monchengladbach as well as featuring for West Germany.
Sancho is reportedly set to sign a new contract with Dortmund paying €210,000 per week, though the 19 year-old has also been linked with Manchester United.
Dortmund's victory means it moves top of the Bundesliga table having earned a pair of wins from its two games.
Defending champion Bayern Munich is five points back ahead of Sunday's (AEST) trip to Schalke.