Ledley scored once in each half, while James Forrest and Dylan McGeouch were also on target and Gary Hooper netted for the first time in six matches as the hosts extended their lead at the top of the table to 21 points.
The Dark Blues thought they had taken the lead after just five minutes when Gary Harkins' free-kick crept in at the near past but an alert linesman spotted that John Baird had strayed offside.
Neil Lennon's side soon reasserted their control and were ahead through Ledley, who captained the team in the absence of the injured Scott Brown.
Dundee keeper Rab Douglas could only palm an Emilio Izaguirre cross into the path of Ledley and he slotted home from eight yards (13).
Gary Irvine then pulled off a crucial block to deny Forrest's goalbound effort before Hooper came to the fore.
First he fired over following some neat link-up play with strike partner Anthony Stokes before Douglas palmed away a stinging effort, with a subsequent low angled drive fizzing just past the upright.
The visitors briefly rallied, with Fraser Forster doing well to tip a deflected Harkins shot over the bar.
The same player was then denied soon afterwards, his header bouncing up off the turf before hitting the crossbar.
That was as good as it got for the Taysiders though as they were reduced to 10-men early in the second half and conceded from the penalty spot.
Matt Lockwood, only on the pitch for three minutes after coming on as a half-time substitute, was dismissed on 48 for bringing down Hooper right in front of goal and Forrest send Douglas the wrong way with his spot-kick.
Stokes then scuffed an inviting shot past the post but Dylan McGeouch killed the game off with a header into the top corner after the impressive Stokes had clipped the ball into the box (57).
Ledley notched up his second goal of the contest when he curled left-footed into the top of the net after Stokes had laid the ball off to him, with Dundee struggling to cope (73).
Celtic continued to press and Hooper ended his mini goal drought in emphatic fashion seven minutes from time, controlling Ledley's ball into the box before volleying home.
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