Russ Wilcox was sacked by Scunthorpe last week after a poor start to the campaign, which had seen the club pick up just two wins in League One following last season's promotion.
With Andy Dawson in temporary charge and Robins watching from the stands, Scunthorpe beat Gillingham 3-0 at the Priestfield Stadium on Saturday.
Robins has now signed a three-year deal, the former Manchester United striker returning to management for the first time since leaving Huddersfield Town in August.
"First and foremost, I'm delighted. I watched the game on Saturday and was very encouraged with what I saw," said the 44-year-old at a media conference on Monday.
"Gillingham is not an easy place to go and we controlled that game.
"Every job is a different challenge but saying that, it's also the same challenge, we have to start picking points up.
"You've got to work hard and make sure the players are capable of doing what you ask of them. If not, that's when we look at changes."
Robins' first game in charge will come on Saturday when Colchester United are the visitors to Glanford Park.
Chairman Peter Swann added: "When I made the decision [to dismiss Wilcox], I had an idea of who I wanted. Mark has a proven track record.
"We both bought into each other. What Mark wants, I want and vice versa.
"It's a three-year deal. There's protection in there for the club and there's protection for Mark as well."
