Argentina striker Higuain, sidelined with a leg problem for the past two matches, scored in between Cutrone's crucial brace amid a game-changing burst of three goals in nine second-half minutes.
Milan had slipped back into old habits prior to their 70th-minute equaliser, struggling to convert possession into chances as their six-match unbeaten run in all competitions appeared destined to end.
But, having fallen behind to Miguel Angel Guerrero's first-half header, the Serie A side rallied upon Cutrone's introduction and overcame a lacklustre opening hour to stay top of Group F on six points.
Milan would have been ahead early had an offside Samu Castillejo not absent-mindedly touched Giacomo Bonaventura's goal-bound header into the net.
Instead, they fell behind from Olympiacos' first real attack in the 14th minute as Leonardo Koutris crossed for Guerrero to expertly nod into the bottom-right corner.
Jose Sa protected the advantage with a low save from Higuain and the Portuguese goalkeeper should have been worked a second time before the break, but Castillejo misjudged Suso's inviting delivery.
The Rossoneri steadily stepped up the pressure after the interval and parity came when Cutrone met Ricardo Rodriguez's cross with a looping header.
Suddenly buoyed, a second came six minutes later through Higuain's cool-headed finish in a crowded box and the startling comeback was quickly capped off by Cutrone, the match-winner tapping home from fellow substitute Hakan Calhanoglu's fizzing low ball across the face.
What it means: Ibra on hold
Speculation in the lead-up to this match suggested Milan are weighing up a surprise move for former favourite Zlatan Ibrahimovic. It was timely, then, that the club's two current strikers both got on the scoresheet at San Siro, perhaps enough to sway the hierarchy's thinking away from the LA Galaxy man.
Milan host Chievo in their final match before the international break, while Olympiacos visit rivals AEK Athens. They both return to Europa League duties on October 25, with Milan entertaining Real Betis and Olympiacos heading to Dudelange.
