They probably recently extracted some viable DNA from the crime scene evidence that had been kept all these years, but they are unlikely to have any DNA taken from the long deceased perpetrator - assuming he was buried, it is unlikely any viable DNA survived 40 years. But if he he still has living relatives, you can take DNA from them and compare it to the crime scene samples, and look for sufficient matches to conclude he was likely to have committed these crimes.
I remember The End of Eternity as being quite unlike anything else Asimov wrote, and all the better for it.