Stewart, a three-time Cup champion, is out indefinitely after suffering a broken right tibia and fibula in an Aug. 5 sprint car crash at Southern Iowa Speedway. He was replaced this past weekend at Watkins Glen by Max Papis, who finished 15th in the No. 14 Chevrolet.
Dillon, who leads the Nationwide standings by three points over Sam Hornish Jr. and five over Regan Smith, was selected in part because of his relationship with Bass Pro Shops, which sponsors him in the Nationwide Series and Stewart in Cup.
"It's a great opportunity for me," said Dillon. "We're really focused on the Nationwide Championship as well, and we're going to give both parties a great opportunity at a win this weekend. We're going to go out there and try to run strong."
Stewart, who was released from a North Carolina hospital on Sunday after having surgery to place a rod in his leg earlier in the week, played a part in the decision.
"I've watched Austin for many years and at each stage of his racing career, he's been successful," Stewart said in a statement from Stewart-Haas Racing. "He's proven to be a very fast learner, and he's able to adapt quickly. Being the youngest champion in the history of the Truck series is proof of that."
Dillon, who will compete in Saturday's Nationwide Race at Mid-Ohio Sports Car Course, recently won the inaugural Truck series race on Stewart's dirt track in Rossburg, Ohio.
"His background in dirt racing is an asset, and off the track we share many of the same interests, specifically, hunting and fishing," Stewart said. "I know he'll perform well at Michigan, and I'm just as confident in his ability to properly represent Mobil 1 and Bass Pro Shops throughout the race weekend."
SHR has not announced who will drive the 14 the following week at Bristol Motor Speedway.
"Well, it's going to be a weekly occurrence here for a little while," said SHR competition director Greg Zipadelli. "We'll evaluate each race as it comes and do the best job we can with putting someone in it we feel will do a good job for Stewart Haas and all of our partners.
"We will probably know more next week. Tony has a doctor's appointment Wednesday here to just kind of go over things, and at that point we'll have a lot better idea of what we're doing. But for right now it will probably be a week by week decision as far as who is in the car.
Sunday's race will be Dillon's 10th start in the Cup series, where he plans to move full time in 2014 for his grandfather's company, Richard Childress Racing -- likely in the No. 3 last driven in Cup by the late Dale Earnhardt in 2001.
The irony is Dillon will be helping SHR, which has signed RCR driver Kevin Harvick and sponsor Budweiser to its organization after this season.
Dillon has two career Cup starts at Michigan, including June, when he finished a career-best 11th.