The Accies, who remain top of the Scottish Premiership table on goal difference, will rue not making the most of their opportunities before the break.
Both goals were crafted by midfielder Gary Harkins and arrived during a five minute spell midway through the second period.
With 68 minutes on the clock, Clarkson powered home a header at the far post after meeting a Harkins corner.
And then Stewart added a second with a left foot finish into the bottom right-hand corner as he latched onto Harkins’s headed assist.
As Hamilton tried to reduce the deficit, Jesus Garcia Tena narrowly failed from a free-kick while Louis Longridge headed over.
During a goalless first 45 minutes, Hamilton created the better chances although Dundee certainly enjoyed their fair share of possession.
Home keeper Scott Bain denied Mickael Antoine-Curier early on while Tony Andreu also went close for Hamilton on 15.
A Dougie Imrie strike from outside the box flew wide with Michael Devlin missing a close range header.
But the hosts responded after the break to secure their first home win of the season and end Hamilton’s long unbeaten run.