The Jets, who have eight players in COVID-19 protocol, were playing their first game in seven days. Winnipeg is 4-2-0 since Dave Lowry replaced Paul Maurice as coach on Dec. 17.
"The resilience of our group was outstanding," Lowry said. "Our penalty kill, our goaltending -- just outstanding. I liked the energy our young guys brought, and our top guys stepped up and played like our best players."
Detroit (16-17-5) has lost four straight and six of seven. Alex Nedeljkovic made 29 saves.
"It's frustrating, because we had a ton of chances, but we didn't take advantage of them," Red Wings captain Dylan Larkin said. "Our process is good, but we have to find a way to turn that into results."
Copp gave the Jets a 1-0 at 5:41 of the first period, banking a shot from behind the goal line in off Red Wings defenseman Nick Leddy's leg. Defenseman Dylan Samberg, making his NHL debut, got an assist.
Copp made the Winnipeg lead 2-0 at 11:08 of the second period, scoring on a breakaway as a Red Wings power play ended.
Winnipeg took two quick penalties to give the Red Wings a 5-on-3 advantage with 4:29 remaining. Red Wings coach Jeff Blashill immediately pulled Nedeljkovic to make it 6-on-3, but Detroit managed one shot in 73 seconds with a three-man advantage.
"That was crazy," Hellebuyck said. "I haven't been in too many of those situations. I knew I was going to have earn my shutout, but the guys in front of me did a phenomenal job."