After a forgettable 90 minutes at the Luigi Ferraris, Cutrone side-footed home from Andrea Conti's teasing centre and followed that up with another fine finish to earn a scarcely deserved win for the Rossoneri in the additional time.
Sampdoria will feel it created the better of the chances against a Milan side that lacked creativity and ideas until Cutrone's introduction, but Pepe Reina made crucial saves from Riccardo Saponara and Albin Ekdal when it mattered.
The win is unlikely to do much to ease the seeming ever-present rumours that Gennaro Gattuso's position as head coach is under threat, although they can at least look forward to a last-eight tie with either Napoli or Sassuolo.
Milan had a let off in the 23rd minute when Jakub Jankto's dipping right-wing cross was spilled by Reina and Gianluca Caprari's claims of a push by Cristian Zapata in his attempts to reach the ball fell on deaf ears.
Gonzalo Higuain wastefully drilled wide of the left post after getting in behind the defence and the lacklustre nature of the match continued until Hakan Calhanoglu wasted a glorious 70th-minute chance from Samu Castillejo's right-wing delivery, somehow squaring across goal instead of turning the ball home at the back post.
Higuain then rightfully had a goal ruled out for offside, and Fabio Quagliarella fired wastefully into the side-netting for Sampdoria before ex-Milan man Riccardo Saponara placed an effort wide of the bottom-left post.
Reina came to the fore in extra-time, first brilliantly closing the angle to deny Saponara before a brilliant double save denied Ekdal and Dawid Kownacki.
Those interventions proved crucial when Conti, whose time at Milan has been injury plagued, put one on a plate for Cutrone to guide past Rafael, and the Italy international lofted another expert finish over the goalkeeper from Calhanoglu's pinpoint pass to seal the win.