Sancho, who revealed a "justice for George Floyd" t-shirt after scoring his first goal in the 57th minute, put Dortmund 2-0 up following Thorgan Hazard's opener.
It took the youngster onto 15 goals for the season, making him the first English player to register at least 15 goals and assists in a single season in one of Europe's top five leagues since the 1994-1995 Premier League campaign, when Matt Le Tissier did so for Southampton.
The winger also became the first player to achieve the feat in a Bundesliga campaign, since the start of detailed data collecting in 2004-2005.
Sancho was not done there, however, and drilled in a second in the 74th minute before rounding off the win - and his hat-trick - in stoppage time.
Dortmund's victory moves them onto 60 points, four clear of Borussia Monchengladbach and Bayer Leverkusen, though seven adrift of Bundesliga leaders Bayern Munich.