A first leg 1-1 draw in the Russian capital left the tie delicately poised for the return in Glasgow, which started amid a cauldron of noise.
Scott McDonald and Jan Vennegoor of Hesselink both threatened with headers in the opening minutes, before Boruc made a smart save to deny Welliton.
With 25 minutes played Spartak should have gone ahead, but Roman Pavluchenko struck a post from the penalty spot after Gary Caldwell had been adjudged to have handled in the area.
Two minutes later the Bhoys were ahead as McDonald pounced to bury his effort past Stipe Pletikosa from 12 yards after a mistake by Radoslav Kovac
However, right on half-time it was all-square again as Pavluchenko made up for his earlier failure by netting following Mozart's corner in from the right.
The second half saw the SPL champions force the pace, but Shunsuke Nakamura and McDonald both spurned chances to secure a place in the group stages.
Six minutes from time Spartak were reduced to ten men as Martin Stranzl was dismissed for a professional foul on Vennegoor of Hesselink, but the full-time whistle came soon after to bring on extra-time.
A tense extra half-an-hour saw few chances for either side, but the Bhoys were handed an opportunity to seal the tie six minutes from time when Kovac handled, but Vennegoor of Hesselink blazed his penalty against the crossbar.
However, the Dutchman was rescued in the shootout as Boruc saved from Egor Titov and Maxim Kalinichenko to ensure a 4-3 success for Gordon Strachan's side.