The hosts, who were looking to rebound from a 2-1 loss to Union Berlin last month, took the lead at the Signal Iduna Park when Swiss centre back Manuel Akanji headed home a Sancho corner in the 66th minute.
"It wasn't an easy match. Wolfsburg started the game well but we found our way into it more after 15 minutes," Akanji said.
"We created a few opportunities for ourselves and might have scored one or two goals. Defensively, we were solid.
"I'm happy that Jadon played in such a good cross for me and I was able to get the goal. We're happy with the three points."
Striker Erling Haaland, who missed Dortmund's previous four league games with a hamstring injury, twice forced spectacular saves from Casteels, the first of which, just before halftime, was scuffed wide on the rebound by Sancho.
Edin Terzic's Dortmund moved to fourth in the league on 25 points, six behind leaders RB Leipzig.
Wolfsburg slipped to sixth after tasting defeat for the second time in three league matches.