Neil Lennon's side endured an uncomfortable evening, drawing 0-0 with Elfsborg in Sweden.
But the SPL outfit came through their third qualifying round tie thanks to a 1-0 win at home in the first leg.
A Roman Shirokov double allied to goals from Hulk, Danny and Andrey Arshavin helped Zenit complete a 6-0 aggregate victory and 5-0 success on the night over FC Nordsjaelland in Russia, while PSV cruised past 10-man Zulte Waregem courtesy of a 5-0 aggregate win.
A Tim Matavz penalty put PSV on their way to a 3-0 win on Wednesday before Zakaria Bakkali and a Bruno Godeau own goal put the game beyond any doubt.
Austria Wien led 1-0 ahead of their trip to FH and did enough to squeeze through with a goalless draw, while a 1-1 first-leg draw allowed Legia Warsaw to make the next round ahead of 10-man Molde by virtue of away goals, with Emmanuel Ekpo dismissed for two yellow cards for Ole Gunnar Solskjaer's side.
Elsewhere in Europe, Viktoria Plzen rounded off a 10-2 aggregate victory over Estonia's Nomme Kalju by thrashing them 6-2 in the Czech Republic thanks to a hat-trick from Pavel Horvath and further goals from Tomas Horava, Tomas Wagner and Stanislav Tecl. Damiano Quintieri netted a brace for Kalju.
Dinamo Zagreb gained an equally comfortable win over Sheriff in Moldova, with Junior Fernandes, El Arbi Hillel Soudani and Duje Cop completing a 4-0 overall win and 3-0 triumph on the night.
And Metalist's 1-1 home draw with PAOK was enough to put them through, having won 2-0 in the reverse fixture. Sebastian Blanco's 72nd-minute opener was duly cancelled out by Tomas Necid 11 minutes later.
