A brace of tries from replacement winger Robert Lyttle - on his competitive debut - and one apiece from Ruan Pienaar, Jacob Stockdale and captain Rob Herring, with two conversions from Pienaar, secured a 29-8 win.
Ulster had found themselves 8-0 down after 10 minutes, but they picked up strongly from the second quarter.
Clarke said: “Really important for us to start well, get a win and build an innings within that and come up with the bonus point victory
“Being 8-0 down, the boys showed a lot of character. There was good leadership from Rob Herring and we took our chances.
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“We really dominated after half-time and got ourselves into a decent position in the second half. It was a spirited performance but we know there is still work to do.”