The Buddies have won just once since lifting the trophy back in March and their miserable run continued as they slumped to a 2-1 second round defeat against Queen of the South.
The match at Palmerston went to extra-time after a goalless 90 minutes but the Buddies looked set to progress after Steven Thompson's header put them in front five minutes into the additional period.
But Stephen McKenna struck a stunning leveller for the Scottish Championship side before Michael Paton clinched victory five minutes from the end of extra-time with his clever lob over David Cornell.
The upset was Saints' third defeat from four matches this season but under-pressure Lennon appealed for unity.
He said: "The supporters invest so much at this club - financially and mentally - so they are right to voice their frustrations.
"But I'm angry and I hope the players are angry. But it is only collectively that we are going to come out of this.
"There are good characters and good players in our changing room. But there seems to be a barrier stopping us from getting over the line. It baffles me."