Sancho rejoined Dortmund earlier this week on a six-month loan after a disappointing stint at United which saw him jettisoned by manager Erik ten Hag.
But the England international set up a goal for Marco Reus only 20 minutes after coming off the bench.
Dortmund were holding on to a 1-0 lead thanks to an early Julian Brandt goal against the Bundesliga's bottom club Darmstadt when Sancho found Reus, who tapped in from close range.
"I feel like it was meant to be, me coming back and seeing Marco again," Sancho, 23, told Sky Germany.
"He's a great friend of mine... I'm happy to set up his goal today."
Youssoufa Moukoko added a third in injury time.
Sancho said he "felt at home" after coming back to Dortmund.
It was Sancho's ninth assist for Reus and his 64th in total for Dortmund.
Sancho first signed for Dortmund as a 17-year-old in 2017 and spent four years at the club, winning the German Cup in 2021 before moving to United for a fee of £73 million.
Sancho had been frozen out at United for most of the season after becoming involved in a public spat with Ten Hag and had not played since August.
"Just to be happy again, to be on the pitch and to help the team," Sancho said when asked what were his goals at Dortmund.
Reus said: "(He's) simply a difference maker... in each game he's good for a goal or assist.
"I love it... Unbelievable to play with him and I'm happy he made his way back to us.
"He hasn't played for seven or eight months, the assist will give him some self-confidence."
Dortmund coach Edin Terzic, who was interim boss in Sancho's final season of his first spell at the German club said: "We're all happy for Jadon.
"What we cannot forget is we are talking (about) Jadon's talent, but he's still a young player, and he's experienced a lot in the past few years.
"He showed us we're not finished with each other."
The win, Dortmund's first in the league since November, took Terzic's side to within three points of fourth-placed RB Leipzig, who lost 1-0 at home to Eintracht Frankfurt earlier on Saturday.
