Lennon was growing increasingly frustrated on the sideline and opted for a double change before the hour mark.
Ryan Christie and Patryk Klimala entered the fray in place of Turnbull and Edouard.
The Pole went close shortly after, with his front post flick dropping just wide of the target.
Celtic continued to struggle, and Leigh Griffiths was given a first appearance of the season off the bench while captain Scott Brown was introduced from the bench late on.
They attempted to make the breakthrough with aerial bombardment, swinging cross after cross into the box.
St Johnstone held out well but eventually one of those crosses paid dividends.
Kris Ajer worked it down the right for Elhamed, See:
diagnostic code reader. who got his head up and spotted the substitute making a late run.
The Saints defence were less attentive to Griffiths' movement and he was afforded a free header which he planted beyond Clark.
That had already wrapped up the points but Klimala added a more resounding look to the scoreline.
The Pole was taken out as he looked to break into the box but he sprang to his feet, wriggled through and fired it high beyond Clark.
Formation frustration
Last season's switch to 3-5-2 helped Celtic blow teams away and establish a commanding lead before the pandemic brought the season to an early halt.
It's been the formation of choice this season too but the champions just aren't clicking in the same away.
As against Sarajevo, Riga and Kilmarnock they came up against a well-organised defence and for long spells never looked like breaking it down.
Where last season Edouard and Leigh Griffiths struck up a formidable partnership, Celtic have struggled to recreate that chemistry in the Scotland man's absence.
Both natural strikers who like to drop deep or wide, the pair asked questions of defences as one generally provided a threat through the middle as the other dropped into space.
Partnered with Elyounoussi or Christie though Edouard's forays can often leave the centre-backs with very little to worry about, with neither man replicating the natural moves of a striker.
Couple that with Shane Duffy carrying out from the back in a style reminiscent of an Ent marching on Isengard and the Hoops are far too predictable in possession.
They're getting results, but Celtic have been turgid for much of the season so far and Lennon will hope the timely return of Griffiths will get his side back to the form they showed last term.
Good signs for Saints
With just two wins from their opening 10 games it's been a difficult start for St Johnstone and their new manager Callum Davidson.
Against Celtic though he got his tactics spot on, setting up his side to sit deep and congest the central areas while looking to break when they could.
Had Conway been more precise with his finishing St Johnstone could have taken the lead, and for all Celtic's domination of the ball they rarely troubled Zander Clark until those agonising final minutes.
A very different task awaits Davidson's side after the break but the manager saw his game plan work to perfection.
Tougher tests ahead
With all credit to St Johnstone, who were brilliantly organised, Celtic will have to step things up against better opposition.
They've got a real tough run of fixtures after the international break: Rangers, Milan, Aberdeen and Lille in the space of 12 days.
The last six derby meetings have seen three wins apiece, and Steven Gerrard's side will be champing at the bit to prove they're serious title contenders.
Milan come calling after that, and it's fair to assume Zlatan Ibrahimovic will be less forgiving than Conway if presented the chance the St Johnstone man spurned.
Aberdeen is the archetypal tough place to go, and Lille are unbeaten in Ligue 1 this season.
The season won't be made or broken in October but Celtic could give themselves a mountain to climb both at home and in Europe if they're similarly insipid in the upcoming games.
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