The 24-year-old German's test was announced by BMW motorsport chief Jens Marquardt following the DTM's season finale at Hockenheim on Sunday.
"It is up to us to reward [Wittmann] for his services," Marquardt said. "Therefore, BMW M GmbH has come up with something special - I am delighted we can make his childhood dream come true with the Formula One test.
"Thanks to good relations with our premium partner Red Bull, we were able to arrange a test drive at Toro Rosso. We are excited to see how he fares in this vehicle!"
Wittmann, who also extended his contract with BMW, told his official website: "I'm overwhelmed. The whole season was amazing; now standing as a DTM champion [with] the contract extension and a Formula One test - this is a dream for me."
Wittmann raced in a variety of single-seater championships prior to his move into saloon cars, and in 2011 finished runner-up to Caterham F1 tester Roberto Merhi in the Formula Three Euro Series.
He switched into the DTM two years later, scoring one podium and finishing eighth overall in his rookie season. He has triumphed four times this year, wrapping up the title with two races to spare.