Lautaro Martinez combined wonderfully with strike partner Romelu Lukaku to score his 15th goal of the season in the fourth minute but Antonio Conte's side were then largely outplayed by a typically ambitious Atalanta.
Martinez was fortunate not to concede a spot-kick when he impeded Rafael Toloi towards the end of the first half, while substitute Ruslan Malinovskiy rattled the upright after the restart.
Robin Gosens poached a deserved equaliser 15 minutes from time and only Handanovic's heroics from Luis Muriel's 88th-minute effort from the spot denied Atalanta the victory.
The Nerazzurri's lethal front two made their presence felt early when Martinez took Lukaku's return pass away from Toloi and blasted under Atalanta goalkeeper Pierluigi Gollini.
In keeping with their vibrant attacking style, Atalanta sought a swift response – Mario Pasalic curling wide from the edge of the area before Duvan Zapata was unable to get enough power on his header to trouble Handanovic.
More textbook forward play from Lukaku and Martinez in the 22nd minute ended with Jose Luis Palomino heading the Argentina striker's chipped attempt off the line.
Martinez almost blotted a fine half when he grabbed at Toloi's ankle as the defender followed up a 41st-minute header saved by Handanovic, although referee Gianluca Rocchi inexplicably decided not to review the incident.
Atalanta boss Gian Piero Gasperini introduced Malinovskiy for Zapata in the 53rd minute and the Ukraine international almost had an instant impact, curling a sweetly struck left-footed effort from Alejandro Gomez's pass against the post from the edge of the area.
Waves of Atalanta attacks continued and they had their reward 15 minutes from time when Josip Ilicic's right-wing cross fell to Gosens for a smart back-post finish.
Alessandro Bastoni clumsily bundled into Malinovskiy during the dying minutes but fellow substitute Muriel fluffed his lines from 12 yards.