The treble treble winners, who have captured the last 10 trophies in Scotland, had to thank attacker Ryan Christie for his whipped-in free-kick in the 81st minute at McDiarmid Park which went past Saints keeper Zander Clark.
It was Celtic's 34th successive victory in domestic cup competitions and they were rewarded with a match against the Dons, with Hearts taking on Edinburgh rivals Hibernian in the other semi-final.
When asked about the relentless mentality of Celtic in cup competitions, Lennon said: "That's 34 undefeated and that's amazing.
"It can go at any time so they want to keep it running as long as they can and they are coming up against all sorts of opposition and they're dealing with it.
"They've faced some difficult opposition – and today was difficult, mentally and physically after Thursday.
"They have picked themselves up again on a really heavy pitch and came through it unscathed, hopefully."
Lennon was referring to Celtic's Europa League last-32 exit at the hands of FC Copenhagen on Thursday night, with the Hoops going out 4-2 on aggregate after losing 3-1 at Parkhead.
The former Celtic captain praised his side for responding positively to the bitter European blow.
He said: "They are good that way. They didn't have much time to dwell on it.
"It was only a couple of minutes in the game on Thursday that they let themselves down, a couple of basic mistakes.
"I didn't think we got what we deserved on Thursday, I thought we were the better team.
"But we got punished for lackadaisical mistakes, that are not like us.
"That's Europe gone for this season and we still have a lot to play for domestically.
"Their fitness levels were good too. Some of them were out on their feet at the end which is understandable because you know what?
"They are human beings and they can't keep churning out result after result and performance after performance.
"People expect that of them but it's not easy.
"There is a good core there and they just will not leave it. There's something in there that they just want to keep it going."
St Johnstone boss Tommy Wright was proud of his side's performance and claimed they did enough to merit a replay.
He said: "We earned enough to get a draw but we didn't defend a set-play and that has cost us.
"We thought we weathered the storm, particularly in the second half when Celtic started it better than us.
"We went toe-to-toe with the best team in the country for a long time and just fell short. Both keepers were excellent in difficult conditions.
"Zander will be disappointed, not for the goal because there is not really a lot he can do but at the end of the day it is a terrible way to go out the competition, off a set-play."