Extra-time appeared probable when Parkhead striker Gary Hooper levelled Rudi Skacel’s opener with four minutes remaining.
However, in the 90th minute Norris dramatically pointed to the spot after a shot from Jambos skipper Marius Zaliukas hit the arm of Joe Ledley and substitute Craig Beattie, a former Parkhead striker, scored to set up the first Hearts versus Hibernian final since 1896.
Lennon, who has three disciplinary cases pending, was further enraged when Norris refused the Clydesdale Bank Premier League champions a penalty in injury time when the ball appeared to come off the arm of Hearts defender Andy Webster.
The former Celtic skipper did not turn up at the post-match press conference but coach Alan Thompson confirmed the Irishman had been into the referees’ room afterwards.
He said: “I think he has been in, yes, but I wasn’t there so I don’t know what was said.
“We are all angry. We have just lost the semi-final of the cup when there were 92 minutes on the clock, it is frustrating.”
Lennon took to Twitter to vent his anger, claiming Norris said the spot-kick was awarded for a handball from Victor Wanyama rather than Ledley.
Lennon said: “Referee told players he thought Wanyama handled...feel so sorry for players and fans..I think it’s personal myself.’’
Thompson was livid about the two penalty incidents.
“I think they are very similar,” said the former Celtic midfielder.
“I’ve seen each three or four times and if he gives one then he has to give the other.”
Hearts will now play Hibernian after Pat Fenlon’s side beat Aberdeen 2-1 in Saturday’s semi-final.
CELTIC: Forster, Lustig (Wanyama 74), Kelvin Wilson, Loovens, Mulgrew, Commons, Brown (McGeough 82), Ki, Ledley, Samaras (Stokes 60), Hooper.
HEARTS: MacDonald, McGowan, Zaliukas, Webster, Grainger, Barr, Driver, Black (Santana 65),Robinson (Beattie 46),Skacel, Elliott (Prychynenko 78).
Ref: Euan Norris (Scotland).