Faced with a target of 309, Derbyshire crumbled for the second time in the match and were bowled out for 117 before tea on the third afternoon with Hannon-Dalby claiming 4 for 24.
The home side had a chance when they bowled Warwickshire out for 155 with Cohen taking 5 for 43 but a post lunch collapse condemned them to a fourth defeat of the season.
Cohen said: "First-class cricket is really tough in general and it has been difficult for the side for the last few weeks. We've just tried to do our best every time but the guys are knackered.
"We've given our all every time and that's the nature of cricket , sometimes it doesn't work out but the guys train hard and it will come off at some point and I'm really hoping it's soon."
Derbyshire were in with a slim chance of getting that first win when they took the last three Warwickshire’s wickets for only 23.
Cohen and Sam Conners both got lavish swing to restrict Warwickshire’s lead to 308 but their hopes quickly faded as Oliver Hannon-Dalby removed both openers in the first five overs.
Billy Godleman was caught at first slip and Luis Reece was caught down the leg side trying to pull.
Craig Miles delivered another body blow with the last ball before lunch when Brook Guest was caught behind off an inside edge.
Wayne Madsen and Ben McDermott briefly lifted the spirits before the wheels came off again with four wickets falling for eight runs in five overs.
Will Rhodes had Madsen caught before Hannon-Dalby returned to pin Matt Critchley lbw.
McDermott went in his next over when he pushed forward and edged low to third slip and Alex Hughes was beaten by a full, swinging delivery from Rhodes.
Spinner Danny Briggs struck with his third ball when Cohen was stumped.
Anuj Dal edged Liam Norwell to third slip, before Ben Aitchison skied Briggs to mid off to end another disappointing contest.