Inter were left frustrated as they mustered 14 attempts to no avail before the break, but the competition debutants' resistance eventually broke when Marcelo Brozovic fired home early in the second half.
Milan Skriniar doubled the Nerazzurri's lead before substitute Alexis Sanchez powered home a third within a minute of his introduction, and Adama Traore headed in a stoppage-time consolation for Sheriff.
Sheriff began their maiden Champions League campaign by sensationally beating Shakhtar and Real Madrid before losing to Inter at San Siro, making this an unexpectedly crucial clash in the battle for a place in the knockout stages.
Inter dominated the first half, boasting 67% possession and having 14 attempts on goal, although only one found the target as they failed to break down the resilient hosts.
Edin Dzeko headed wide from a corner and later had an effort well saved by Giorgios Athanasiadis, while Lautaro Martinez came close five minutes before the interval when his shot from the edge of the area struck the post.
Sheriff's resistance broke in the 54th minute when Brozovic feigned a shot with his left foot at the edge of the area before dragging the ball back to his favoured right and shooting into the bottom corner.
Inter's corner kicks had caused problems for the Sheriff defence and were the source of the second goal, with Stefan de Vrij heading a delivery straight at Athanasiadis before Skriniar tucked in the rebound at the second time of asking.
Sanchez needed less than a minute to make his mark after being introduced on the 82nd minute, thumping a powerful strike into the roof of the net for his first Champions League goal since February, 2017.
The home crowd, who had little cause for excitement all evening, were finally given something to cheer at the death when Traore headed home a Sebastien Thill cross.