United had been forced to scrape past Ross County on penalties in the last round but never really looked in danger at New Douglas Park.
The holders scored after 20 minutes when Craig Conway's quick free-kick was collected by Paul Dixon, whose shot was blocked by Tomas Cerny and fell for Jon Daly to send a firm header into the back of the net.
United found the back of the net again after 39 minutes when Dixon tried his luck with a shot from the edge of the box that deflected off Martin Canning past the grasp of Cerny and into the bottom corner.
Hamilton hauled themselves back into the tie less than a minute after the restart when Andy Graham's cross from the right found Mickael Antoine-Curier at the back post and he bulleted a header past Dusan Pernis.
But Dundee were on target again with 58 minutes gone. Jon Routledge fouled David Goodwillie and Conway flighted the resultant free-kick into the box for Daly, who nodded across goal for Prince Buaben to stab home from close range.
Source: PA
Source: PA