Incorrect. Somebody can leave behind their DNA on a cigarette butt, a napkin, some fallen hairs. The reason why DNA analysis is connected with serious felonies is that it's mainly in serious felonies that authorities can justify the expense of going for DNA analysis.
a) they had the perp’s DNA (from a rape or murder IIRC)
b) they had a suspect who they thought might match the DNA
c) the cop followed the suspect and grabbed his littered cigarette butt
d) they matched the cigarette butt DNA to the crime scene DNA.
I guess if a bank robber smoked a cigarette and left the butt behind it might well be used as well; but it would only “work” if the guy's DNA was already in the system, otherwise all you have is DNA with no identity to match it to.
What scenario are you envisioning whereby a non violent criminal gets identified by DNA? They find his DNA where exactly? What type of crime scene?