It was the fourth straight defeat and the worst start to a season for big spenders Schalke, leaving the runners-up of last season anchored at the bottom of the table.
They finished the game with 10 men after a second booking for defender Nicolas Plestan.
Japan's Kagawa, one of the Bundesliga's revelations after Borussia bought him for 350,000 euros from the Japanese second division, struck from close range in the 20th and 58th minutes.
"This is my worst day at Schalke," coach Felix Magath told reporters. "I have never experienced such a start.
"The team is insecure and the defeats have taken their toll. I have never seen the team like that."
With Schalke's defence in disarray, Dortmund could have added another just before halftime when Kevin Grosskreutz hit the bar with a cross and Subotic twice saw his rebounds saved.
Kagawa took advantage of Schalke's poor defence to tap in a Jakub Blaszczykowski cross in the 58th minute for his third goal of the season.
Schalke's big-name strikers, Dutchman Klaas-Jan Huntelaar and Raul, were largely shut out with the Spaniard choking under pressure from defender Mats Hummels.
Lewandowski headed in Dortmund's third in the 86th before Huntelaar struck three minutes later to cut the deficit with his first goal of the season.
