The hosts had a lead of 127 with five wickets remaining going into the final day, but Eckersley's 104 saw them declare 313 in front on 294-9.
It left Derbyshire needing to negotiate 38 overs and they slumped to 43-4, with Billy Godleman out for a golden duck.
Wayne Madsen (42 not out) and Shiv Thakor (31 not out) made the game safe as Derbyshire closed on 104-4.
Leicestershire move up to third in Division Two, 11 points behind leaders Essex, while Derbyshire are still without a win this season.
A second century of the game by Ned Eckersley helped Leicestershire secure a draw with Derbyshire at Grace Road.