Dortmund were without striking Erling Haaland, who was left out of the squad after picking up a minor injury against Bayern Munich on Tuesday. Sancho was handed his first start by Lucien Favre since the Bundesliga returned to action.
An uneventful opening 45 minutes ended scoreless, but the visitors came to life early in the second half with Emre Can forcing a close-range save from Paderborn's Leopold Zingerle and Thorgan Hazard flashing a shot just wide of the far post from a Sancho cross.
Dortmund's second-half pressure finally told when Hazard tucked away a loose ball right in front of goal to make it 1-0 after a Can cross was palmed away by Zingerle.
Sancho then doubled Dortmund's lead minutes later, tapping into an empty net from a Julian Brandt cross before taking off his jersey to reveal a shirt underneath that read "justice for George Floyd" to honour the American black man who died in police custody in Minneapolis on Tuesday.
Dortmund and Paderborn then traded back-to-back goals as Uwe Hünemeier converted from the penalty spot following a Can handball and Sancho buried a shot from inside the area to restore his team's two-goal lead.
Achraf Hakimi and Marcel Schmelzer scored a pair of well-taken goals before Sancho added his third in second-half stoppage time with an unstoppable shot on a Dortmund counter-attack to reach the final scoreline.
Hakimi also lifted his jersey after scoring to show an undershirt with a message of support of justice for Floyd.
The hat trick was the first of Sancho's career and put him on 17 goals for the season at Dortmund to go along with 16 assists for the prolific 20-year-old.
The result sees Dortmund move up to 60 points to remain in second place in the Bundesliga, seven adrift of leaders Bayern, while Paderborn remain mired in last place in the table on 19 points.
Dortmund, the 1997 Champions League winners who last won the German title in 2012, are at home to ninth-placed Hertha Berlin in the next round on Saturday while Paderborn visit fifth-placed RB Leipzig.