Teenage stopper Josh Thompson's first goal for the Hoops gave Tony Mowbray's side a half-time advantage before Saints keeper Graeme Smith conceded a penalty in the 67th minute for fouling Marc-Antoine Fortune.
The Republic of Ireland star, who was always a threat, slammed in the spot-kick to take his tally to nine in nine games since arriving from Tottenham in January.
Substitute Georgios Samaras, on for Fortune, completed the scoring with three minutes remaining as the Parkhead men moved 10 points behind SPL leaders Rangers, although they have played a game more.
Mowbray would have been pleased with his team's performance although omens were on their side having scored five then four against the Perth men in the previous two meetings this season.
In essence, Celtic started with a 4-2-4 formation but despite their injury problems the Perth men were equally positive in their approach.
It was clear early on that Keane would test the Saints' rearguard but in the 12th minute it was Fortune who escaped down the right to latch on to a Paul McGowan pass.
However, the former Nancy striker's cross evaded Keane and the ball sped to safety.
The injury-ravaged visitors suffered a blow in the 15th minute when striker Kenny Deuchar limped off to be replaced by Steven Reynolds.
But that was simply the first part of a double-whammy.
Seconds later Edson Braafheid's in-swinging free-kick from wide on the right was met by Thompson who, among a crowd of players, headed past Smith from eight yards.
It was assumed by the home fans that Celtic would go from strength to strength but there were occasional, and rather curious moments of nervousness when the ball arrived in their box which offered hope to St Johnstone.
But in the 28th minute, in a moment of assertiveness, McGowan worked a one-two with Keane then skimmed the bar with an effort from the edge of the box.
The Perth men slowly came out of their shell and 10 minutes from the break Celtic's stand-in skipper Darren O'Dea put in a good block to prevent Gavin Swankie's shot giving Zaluska too much bother.
Moments later, McGowan headed Braafheid's curling cross past the post when it looked like he ought to have at least hit the target.
In injury time, Keane stretched to get on the end of an Aiden McGeady cross and the ball spun over.
While the Saints had worked hard in the first half to keep themselves in touch, it appeared an uphill task for them after the break as Celtic took immediate control.
However, in the 52nd minute, after Thompson had barged Davidson 25 yards from goal to concede a foul, Swankie curled a free-kick past the far post.
There was audible relief among the nervy Hoops supporters who came to life moments later when, from just inside the St Johnstone box, McGeady curled a shot past Smith only to see it rebound off the post and back into play.
It was a grateful Perth side who eventually emerged unscathed from the passage of play and they had more luck moments later when Fortune missed a sitter with his head after Keane had stood up a pin-point cross at the back post.
As Celtic stepped up the pressure, McGeady curled another effort inches over the bar before Smith made a save from Keane.
Smith then made an even better save from Keane when he blocked the Irishman at close range after a great cross from McGeady.
However, in the 67th minute the St Johnstone keeper was beaten to the ball by Fortune and as the Celtic striker went tumbling, referee Calum Murray pointed to the spot, allowing Keane to blast in his ninth goal in nine games since arriving on loan from Spurs in January.
Three minutes from time, Samaras, on for Fortune, made it 3-0 when he planted the ball behind Smith from 14 yards.