The Reds briefly silenced the home fans in the opening moments when Steven Gerrard punished Ryan Shawcross' shirt-tug on Luis Suarez from the penalty spot, but that was as good as it got for the visitors.
The equaliser came quickly through Jon Walters' assured finish, and the Republic of Ireland striker added his second and Stoke's third in some style after Kenwyne Jones had nodded the Potters ahead.
Rodgers told Sky Sports: "It was a great start for us, getting our first penalty. Coming away from home and starting like that was great, but we gave away poor goals. All three of them were actually quite straightforward to defend.
"The first one, Martin Skrtel slipped and was unfortunate but there were two goals where we have to do much better with and that was the disappointment really.
"We got into some good areas but when you come to places like here, you've got to defend better or else you get nothing and we deserved nothing because of that.
"Everyone knows the strengths of Stoke and they do it very well, but when you've prepared to deal with it's even more disappointing. It was a corner, a throw-in and a direct ball.
"Stoke don't quite get the credit they deserve. Tony Pulis has done a brilliant job here to keep them progressing. They do it very well. But from our perspective, it was poor defending."