It was the first time the Istanbul club have claimed the league title since the 2018-19 season.
Icardi gave Galatasaray the lead in the seventh minute but Ankaragucu striker Milson levelled nine minutes later, before Icardi restored their advantage five minutes before half time.
Galatasaray's Yilmaz scored in the 73rd minute and Sergio Oliveira sealed the win in 78th.
The victory lifted Galatasaray onto 82 points, five points ahead of their second-placed rivals Fenerbahce.
Fans of the club paraded through the streets of Istanbul in cars, sounding horns and waving flags in Galatasaray's red and yellow colours.
Galatasaray and Fenerbahce are set to play against each other in their final match of the season on Sunday.
"It's quite an emotional moment for me, unbelievably great happiness. I would like to thank the great Galatasaray fans, and my players. Hopefully, we will complete the league by winning the derby match next week," Galatasaray coach Okan Buruk told broadcaster BeIN Sports.
The race for the championship had intensified in recent weeks, with Galatasaray in consistently impressive form throughout the season.
Led by Argentinian forward Icardi and Belgian Dries Mertens (36), the team thrilled fans with their attacking prowess and defensive solidity.
The other of the big three Istanbul clubs, Besiktas, are third on 74 points, while last season's champions, Trabzonspor, are in fifth place after Tuesday's games.
Two Superlig clubs, Hatayspor and Gaziantep, withdrew from this year's championship as a result of the devastation caused by earthquakes in southern Turkey in February in which more than 50,000 people died.
