Bildt, accompanied by U.N. Secretary-General Ban Ki-moon, stayed at the U.S. military installation at Bagram north of the Afghan capital before traveling to central Kabul for the international gathering on the future of the country, Swedish news agency Tidningarnas Telegrambyra reported Tuesday.
Bildt said on his blog the plane was about a mile from the runway when it was diverted.
"It naturally created quite an uncertain situation," he wrote.
Bildt said he and Ban had to wait several hours at Bagram before heading to the Kabul conference.
"After the night's drama, (U.N.) Secretary-General Ban Ki-moon and I -- and our colleagues -- finally at (4 a.m.) were able to descend in darkness in two helicopters ... in central Kabul," Bildt wrote.
Danish Foreign Minister Lene Espersen also was forced to delay his arrival in Kabul because of the attack, TT said.
Swedish Foreign Ministry spokesman Anders Jorle said he couldn't say whether anyone was hurt in the attack on the airport or who was responsible.