The 34-year-old had spurned two magnificent opportunities before succeeding with his third to sink the La Liga outfit, who defended doggedly throughout the majority of the match but offered very little threat in the final third.
Prior to kickoff at San Siro, a tribute was held in memory of late Inter wing-back Andreas Brehme, scorer of the winning goal for West Germany in the 1990 World Cup final, following his death at the age of 63.
Simone Inzaghi's side quickly took control of proceedings in the opening exchanges, but Diego Simeone's Rojiblancos defenders were often in the right place at the right time to mop up any loose balls and did not allow the hosts a single shot on target in the first 20 minutes.
Half-hearted appeals for a penalty against Nahuel Molina's handball in the 13th minute were quickly waved away - the defender's arm was by his side as Henrikh Mkhitaryan's strike deflected onto his body via the foot of Axel Witsel.
Simeone's backs-against-the-wall approach continued to win out as the clock ticked down to half time, forcing Lautaro Martinez to try his luck from range in the 32nd minute, but his attempt was high and not at all handsome.
The Argentine would belatedly fire the game's first shot on target in the 36th minute, getting across Jose Gimenez to meet a Nicolo Barella cross, but his header was too close to Jan Oblak.
Just two moments later, Rodrigo De Paul almost negated Atletico's defensive efforts with a terrible pass straight to Marcus Thuram, who beat Witsel to the loose ball and found Martinez in a central position, but the visitors managed to get bodies back to deflect the ball behind for a corner.
That set-piece came to nothing, although Inter were beginning to benefit from some sloppy play from Simeone's side, who needed Oblak to be alert to gather Thuram's low strike in the 43rd minute.
Both managers opted for a slight tweak for the second half, as Thuram made way for Arnautovic in the Inter attack, while Stefan Savic replaced Gimenez in the Atletico rearguard.
A golden opportunity fell to Arnautovic just four minutes into the second 45, as Federico Dimarco delivered a fizzing ball to the back post, but the Austrian was on the stretch and blazed the ball over the top.
Simeone soon bit the bullet and introduced Alvaro Morata, who was named on the bench owing to a knee injury, but the gilt-edged chances continued to fall Inter's way after a brief period of promise from Los Rojblancos.
Arnautovic again found himself in a wonderful position to score in the 63rd minute, playing a slick one-two with Martinez on the edge of the box, but the ex-Stoke City man once again cleared the crossbar from only a few yards out and cut a particularly pained figure in the aftermath.
Inter continued to huff and puff, and Martinez got his head to another cross in the 77th minute - this time from Hakan Calhanoglu - but again, his effort was straight into the gloves of a seemingly impenetrable Oblak.
However, the Slovenian would finally be beaten in the 79th minute, as a mistake from Reinildo saw Martinez race through on goal, and while Oblak was equal to his effort, the hitherto wasteful Arnautovic fired home the rebound with the help of a deflection off of Samuel Lino.
The Nerazzurri's opener was undeniably deserved, but Lino came close to making amends with a venomous 20-yard effort just two minutes after the deadlock was broken; his effort flew a whisker wide of Yann Sommer's goal.
Atletico's attacking verve had eluded them throughout the match, though, and Inter will take a warranted advantage to the Wanda Metropolitano for the return fixture in three weeks' time.
Atletico now prepare to travel to Almeria in La Liga action this Saturday, while Inter make the journey to Lecce in Serie A action on February 25.