Despite being a man down, Dungannon almost extended their advantage in the tie when Cahal McGinty's header from Gael Bigirimana's free-kick went just over the top of the crossbar.
However, just like during the opening 45 minutes, Vaduz dominated the second half and piled on the pressure.
Stopper Dunne continued to frustrate the away side though and gathered Monsberger's header before Danny Wallace cleared Denis Simani's free header away from danger.
Manager Marc Schneider made multiple changes for Vaduz in quick succession, and they nearly made an instant impact when Jonathan De Donno's header crashed off the crossbar.
The away side took the lead on the night and levelled the tie in the 79th minute when substitute Hammerich's low shot from the edge of the box found the back of the net.
Both sides pushed for a winner in normal time when Adam Glenny's effort was kept out by Leon Schaffran and Mischa Eberhard fired just wide at the other end.
During extra time, Javier Navarro's header found the back of the net, but the substitute was offside so the effort didn't stand.
FC Vaduz took the lead on 107 minutes with the unmarked Eberhard found the bottom corner from just outside the box.
Navarro was then denied by the woodwork before Vaduz booked their place in the third round when Hasler scored into an empty net after Dungannon and goalkeeper Dunne had been caught on the break when searching for an equaliser.