The Swansea midfielder was at Anfield on Thursday night to watch the Reds' 3-0 Europa League win over FC Gomel and passed a medical on Friday to seal his switch to the Merseyside giants.
Rodgers sees the signing as another significant step towards building the type of team he wants at Anfield. His first signing, Fabio Borini, got off to the best possible start by marking his home debut with a first-half goal to set Liverpool on their way to an easy victory that sealed their place in today's final qualifying round draw.
"It is important for any striker to score because they are judged on their goals," said Rodgers. "But I brought him here because I know he can score.
"He is only 21 and has a lot of improvement left in him. But with his work-rate and his ability on the ball I think he will fit in very well to our team."
Skipper Steven Gerrard grabbed a second, with Luis Suarez again the architect. The Uruguayan turned in a superb display, showing why Rodgers was so delighted the South American signed a long-term contract extension earlier this week.
"Luis is an incredible player," said the Liverpool boss. "He has come back to us and looked great in training.
"He is so creative in the box but he also presses very well when he doesn't have the ball."
Glen Johnson wrapped up the scoring in a game in which Manchester City target Daniel Agger completed the full 90 minutes.