Motherwell had chances to take the lead at Dens Park before Lee Ashcroft's 78th-minute winner, most notably when Connor Shields hit the crossbar in the first half.
The South Lanarkshire side struggled to make it out of the group stages prior to this second-round exit and are yet to win in the cinch Premiership, losing 3-2 Hibs and drawing with 1-1 with St Johnstone.
Motherwell lost key players such as midfielder Allan Campbell, striker Devante Cole and Scotland defender Declan Gallagher in the summer and are in need of a rebuild.
However, the travelling fans made their displeasure known at full-time, but Graham Alexander has asked the supporters to stick behind his new-look team.
He said: "We are out of the cup and we didn't play to the levels we expect and work for.
"We were pretty good in the first half without taking the opportunities we got but we lost our way a bit and fell to a set-piece. Losing the game to that is really disappointing.
"We want to win games every single week but we know the support gets disappointed when we get beat.
"I wouldn't expect anything different but it won't change the way I approach training for next week's game.
"The players need that backing, I need that backing, we need that support together, it is not just one way."