Protecting a 3-1 advantage from last week's home match, Kashima did enough to reach the last eight for the first time in a decade despite a 2-1 loss in Wednesday's return leg.
The Japanese side made a shaky start as Gen Shoji's error allowed Hulk to net the opener in just the seventh minute, although their nerves were eased by a precious Shoma Doi equaliser three minutes from half-time.
Shanghai was back within a goal when Hulk converted his second from the spot in the 81st minute - Shoji having been penalised for a handball - but the comeback came too late for Vitor Pereira's men as their wait for continental success continues.
Kashima is joined in the next round by Suwon Bluewings, which overturned a single-goal first-leg deficit against Ulsan Hyundai to advance 3-1 on aggregate.
Kim Gun-hee was the hero with two goals in the space of five first-half minutes, quickly following up his 26th-minute header by skilfully controlling and firing home Waguininho's knock-down.
Mislav Orsic had a golden chance to put Ulsan back in the driving seat when Kwak Kwang-seon shoved Richard Windbichler to the floor, but the Croatian striker's low penalty was pushed away by Shin Hwa-yong.
It was the last chance offered by Suwon as Waguininho grabbed one of his own in stoppage time, putting the seal on the two-time champion's impressive 3-0 win.