The match was goalless after 90 minutes before substitute Daniel McBreen headed Saints ahead, however, the visitors' dreams of a cup final were dashed when Nacho Novo equalised from the spot.
Saves by Rangers keeper Neil Alexander in the penalty shootout to deny Steve Milne and Jody Morris allowed Cousin the chance to win the match for the Gers. Alan Main's only spot-kick save was from Brahim Hemdani.
Extra-time was probably the last thing Rangers wanted with their UEFA Cup and SPL commitments. Watching Chris Burke and Steven Naismith stretchered off in the opening 20 minutes was also something Rangers boss Walter Smith did not need with a squad already stretched by injuries and suspensions.
In a poor first half Main did well to block Novo's shot after Goran Stanic was short with a pass-back.
The second half was a great improvement with both teams having chances. Milne's flick at the near post from Liam Craig's free-kick was well saved by Alexander.
Main saved bravely at the feet of Steven Whittaker and also produced some inspired saves to deny Cousin and Novo. Goal-line clearances by Craig and Morris in the last five minutes led to extra-time.
Saints went ahead in the first period when McBreen headed home Paul Sheerin's back-post cross.
When Cousin was brought down by Gary Irvine, Rangers drew level through Novo from the resulting penalty.
Novo hit the post in the second period of extra-time before Rangers edged through after a dramatic shootout.