Second-half goals from Danny Ings and Pierre-Emile Hojbjerg secured the three points for Saints.
Saints manager Mark Hughes said: "It's really pleasing, the level of control we had. Certainly in the first half I thought we were excellent, I thought we were quicker, sharper and more dynamic in everything that we did in comparison to Palace."
Saints also saw Charlie Austin have a penalty saved.
But Hughes insisted: "Obviously, they were the home team and they wanted to start brightly, but they weren't able to get a foothold in the game because of our good play.
"The only negative really going into half-time was that we hadn't capitalised on the dominance we had. We really should have been one or two goals to the good, but it wasn't to be.
"So you do have that slight fear that maybe you might get caught out because you haven't capitalised on that pressure, but thankfully we were able to get our noses in front."