The Hoops stand-in skipper opened the scoring in the 13th minute and James Forrest notched the second from the spot four minutes after the interval when Dees substitute Matt Lockwood, only on for the start of the second half as replacement for Brian Easton, was shown a red card for denying Gary Hooper a clear goalscoring opportunity.
Dylan McGeouch, Ledley and Hooper then added goals to further deflate the relegation-bound visitors, whose new interim boss John Brown watched from the directors' box.
Dundee remain 15 points adrift of St Mirren at the foot of the table with their fate almost inevitable, although there may still be trouble looming in the wake of Brown's appointment.
Disgruntled fans had walked out of a meeting with the Dundee board at Dens Park yesterday after it was confirmed the 51-year-old former Dark Blues player would be taking over following the shock sacking of Barry Smith last week.
Brown might have had second thoughts after witnessing his new charges capitulate after the interval, albeit against a Celtic side who showed no mercy as they stride towards the title.
As expected Celtic boss Neil Lennon shuffled his squad, making five changes in total.
In came McGeouch, Forrest, Thomas Rogne, Beram Kayal and Anthony Stokes with Adam Matthews, Kris Commons, Charlie Mulgrew, Lassad Nouioui and Victor Wanyama dropping out, the latter two starting on the bench.
The Dees showed one change with loan-player Lewis Toshney unable to play against his parent club, replaced by Kyle Benedictus.
There was a moment of controversy in the sixth minute when Dundee forward Gary Harkins' curling free-kick from wide on the left beat Fraser Forster down low at the near post, but referee Kevin Clancy ruled the 'goal' out, presumably for offside.
Celtic appeared to treat that let-off as a sign of insubordination from the visitors and before long Ledley had fired the home side ahead.
Emilio Izaguirre's cross from the left was parried by 40-year-old former Celtic keeper Robert Douglas straight to the Wales international, and he took advantage of his good fortune to quickly knock the ball into the net from 12 yards.
To their credit, Dundee kept going forward and in the 25th minute they had a decent chance when James McAlister's corner from the left was met by Declan Gallagher but the former Celtic youth player headed wide.
A prolonged period of Celtic dominance followed.
Kayal and Hooper both missed the target with efforts but in the 29th minute Douglas made a good save from the former Scunthorpe striker's powerful drive, before making an easier save from Izaguirre's low drive.
As the champions turned the screw, Hooper then drove wide of the far post after being set up by Forrest.
However, Forster was called into action 10 minutes from the break when he made a fine finger-tip save from Harkins' chip from the edge of the box after the Hoops defence had made a hash of clearing a Nicky Riley cut-back.
From Ryan Conroy's resultant corner, Harkins' header beat Forster only to crash off the bar and Celtic breathed again.
Wanyama replaced Efe Ambrose four minutes from the interval as Dundee ended the first half in a positive fashion, although that bullishness disappeared soon after the restart.
After Stokes had the ball in the net only to be penalised for an infringement, Lockwood had to pull back Hooper to prevent the Celtic striker tapping a Forrest cross into an empty net from six yards out.
There was little surprise and few complaints that the Dundee substitute was shown a straight red card by Clancy before Forrest guided the penalty into the corner of the net.
With the Taysiders down to 10 men the traffic was all Douglas-bound, and when McGeouch confidently headed in a Stokes cross for number three, it looked like Celtic could reach the six they managed against Dundee United last weekend.
Stokes came close with a couple of attempts before Hooper forced another save from Douglas with a low drive from 14 yards.
However, in the 68th minute it was the former Cardiff midfielder who again showed how it should be done when he smashed a pass from Stokes high past Douglas from 14 yards.
Ledley turned provider in the 82nd minute with a wonderful lofted pass which Hooper took on his chest before drilling low past the hapless Douglas, who was glad to leave his former stomping ground without further punishment.