Critchley had Colin Ackermann caught at first slip by a diving Wayne Madsen, Ben Mike caught at second slip by Leus du Plooy and Dieter Klein caught behind by wicket-keeper Harvey Hosein in the space of five balls as the Foxes collapsed from 104-2 to 140-8.
Only a defiant last wicket stand of 52 between Harry Swindells and Chris Wright prevented Leicestershire being beaten by an innings.
The morning had seen a double wicket maiden from first-class debutant Ed Barnes raise Derbyshire’s hope of forcing a three day win.
The young Yorkshireman, on loan at Derbyshire until the end of the season, bowled nightwatchman Callum Parkinson off the inside edge as the batsman tried to leave a delivery just outside off-stump, and three balls later found the edge of left-hander Ben Slater’s bat and saw Madsen hold a waist-high catch at first slip to leave Leicestershire struggling on 34-2.
They would have been in even deeper trouble if Leus du Plooy had been able to hold a chance given by Harry Dearden soon afterwards, the batsman edging an attempted cut at a wide delivery from Dustin Melton to second slip.
Dearden played aggressively after lunch, hitting 21 runs off 14 balls and raising a 50 partnership with opener Hassan Azad, but an attempt to clear mid-on off the bowling of Critchley gave Luis Reece a simple catch.
Azad followed, leg before wicket on the back foot to the occasional left-arm spin of du Plooy, and the writing was on the wall for the Foxes when George Rhodes tried to clip a full straight ball from Dustin Melton through midwicket, missed, and was palpably leg before wicket.
Critchley’s three in five made the win all but certain, though Will Davis, Swindells – who finished on a career-best 41 not out – and Wright battled impressively.
With the pitch offering increasing turn, left-arms spinner Parkinson then had Billy Godleman caught behind before Reece and Madsen saw their side over the line to make it two victories from two.