Scott Sinclair ended their 374-minute wait by slotting home from the spot in the 14th minute, after a needless Paul Scharner challenge on Joe Allen.
Leroy Lita grabbed his first goal for the club 10 minutes later, nodding home from Sinclair's header across the box, before Nathan Dyer raced on to a flick-on from the striker to fire in the third shortly after the break.
Rodgers said: "I thought we thoroughly deserved the win, it was a very complete performance. Our performance level has been excellent in the first five games, and that's three clean sheets now, and we have always looked like scoring.
"But we have never had that bit of good fortune, but our play with and without the ball was top class.
"Sometimes if you play like we have been playing and you don't pick up a win it may drag on and you can lose confidence, but it was an outstanding team performance."
Rodgers felt the early penalty was that bit of luck his side had been lacking, and he praised Sinclair's composure as the hat-trick hero of their play-off success fired low to Ben Foster's left.
"It helped us to get the early goal we have been wanting," he said. "It was a terrific run into the box by Joe Allen and Scott's calmness in those situations is incredible. We just needed a break, that was the one that did it for us."