Romelu Lukaku gave Inter a fifth-minute lead from the penalty spot – Diego Carlos might have been sent off for bringing him down - but Sevilla's semi-final hero Luuk de Jong overturned the deficit with a pair of fine first-half headers.
Diego Godin's 35th-minute leveller had Inter swiftly back on terms and the Serie A side appeared to be in the ascendency during a more circumspect second half, until a stunning twist in the tale.
Diego Carlos had been tormented by Lukaku for most of the game but his overhead-kick when Ever Banega's 74th-minute free-kick was partially cleared took a decisive touch off Inter's top scorer and flew into the net.