But the hosts fought back with two goals in quick succession after the break, Charlie Adam slotting in the first in the 58th minute and Shawcross smashing home the second four minutes later.
With Stoke having lost their Barclays Premier League campaign-opener 1-0 at Liverpool last weekend, this result brought Hughes - appointed as the Staffordshire outfit's manager over the summer - not only a winning start at home, but his first points in the job.
And following the victory, Hughes said: "Premier League points are like gold dust and this was important for us - I'm a new manager working really hard with a new group of players.
"In fairness to them, they have worked their socks off to try to implement what we are trying to do.
"We will get better - we weren't great and we can play better, and I think we all accept that. But the important thing was to get a positive result."
Giving his assessment of the contest, Hughes said: "We were a little bit apprehensive in the first half. Credit to Palace, they pegged us back.
"It was a little bit difficult with the wind in our face to get out of our own half in that first half, and we were a little bit too respectful of Palace and didn't get close enough to them.
"They were able to put passes in between the lines and that caused us a problem initially, although they didn't create too many chances - we created one for them, which was uncharacteristic from Ryan.