After a goalless first half, Kane broke the deadlock just seven minutes after the restart. Leon Goretzka brought the ball down outside the box and passed to the England striker, who took his time before picking out the bottom corner.
Inter were level within six minutes when the German side failed to deal with a corner, allowing Lautaro Martinez to fire beyond Jonas Urbig. Inter's comeback was complete just three minutes later as Hakan Calhanoglu's corner found Benjamin Pavard to head into the net.
Vincent Kompany's men were given a glimmer of hope when Leroy Sane picked up a poor clearance and sent an inviting ball towards the back post, where Eric Dier sent it looping over the line to make it 2-2.
Thomas Muller looked capable of a spectacular finish as he diverted a header towards goal deep into stoppage time, but Yann Sommer stopped it as Inter booked their place in the semi-finals with a 4-3 aggregate win.