The operation took four hours. IS continued to fire mortars at the airport from further inside the city after losing the ground to the army.
The jihadists have also entered a nearby military base amid further clashes, a military spokesman said.
Eastern Mosul was retaken last month.
Thursday's advance brings the army very close to Mosul's south-western outskirts, where the militants are expected to launch attacks from densely populated neighbourhoods.
Aram Shakaram, the country deputy director for Save the Children in Iraq, told the BBC he believed relatively few people had been able to escape the city since yesterday.
The assault began with overnight air strikes by the US-led coalition before armoured columns advanced to the airport's perimeter.
An Iraqi lieutenant was killed by one of the roadside bombs planted in the area by IS, the BBC has learned.
Iraqi forces also came under fire from IS militants who had holed up inside airport buildings, reports said.
Foreign troops from the US-led coalition were with the attacking troops, officials told the Associated Press news agency, without specifying their nationality.
The airport's runway has been destroyed by IS but BBC Middle East correspondent Quentin Sommerville, who is embedded with Iraqi federal police units, says it still has value.
It is a large piece of land and controlling it will help secure southern routes to west Mosul, our correspondent says.
The airport and the al-Ghazlani base are on Mosul's southern outskirts on the western side of the Tigris river.
Thousands of Iraqi troops, backed by artillery and air power, are involved in the assault to retake Mosul.
Leaflets warning residents of an imminent offensive were earlier dropped over western Mosul, where military officials say narrow winding streets could make retaking the area particularly difficult.
Although slightly smaller than the east, western Mosul is more densely populated and includes districts seen as pro-IS.
Trapped in the city
The UN has voiced concern about the welfare of civilians trapped in the city.
Save the Children in Iraq's country deputy director said the charity believed that nearly 800,000 people were trapped in Mosul.
More than 160,000 people have fled their homes in and around the city.
The UN said in late January that almost half of all the casualties in Mosul were civilians.
All bridges linking the east and west of the city, across the Tigris river, have been destroyed by air strikes.
IS jihadists overran Mosul as they spread across much of northern and western Iraq in 2014.