Verlander pitched 6 1/3 innings, striking out at least one hitter in each inning he pitched.
"He was just in control today," Kinsler said of Verlander. "He looked really sharp."
The Tigers jumped on White Sox starter Jose Quintana for five runs in the second inning.
With two outs in the inning, Castellanos drilled a 387-foot, two-run homer to extend the lead to 5-1.
Although the inning was the only one in which Quintana struggled, it proved to be all the Tigers needed.
"I tried to keep the game close," Quintana said. "But the five runs, that's too much against Verlander. He threw the ball well."
Kinsler, who reached base four times, made it 6-1 with a solo shot off of Quintana in the fourth inning.
Verlander registered his final strikeout in the seventh inning when he struck out Avisail Garcia to become the first Tigers pitcher since Mickey Lolich in 1970 to strike out 10 hitters on Opening Day.
"It felt good," Verlander said. "It's good to get the first one out of the way and it's good to get the win."