Italian forward Borini has found first-team chances few and far between at Anfield since signing from AS Roma in 2012 and was sent out on loan to Sunderland last season.
He was deemed surplus to requirements on his return to Liverpool, who accepted bids for the player from both Sunderland and Queens Park Rangers.
The Italian, however, shunned the opportunity of a fresh start at a new club, where his chances of first-team football would have been greatly improved.
After making only one Premier League start in nine games this season, the 23-year-old's chance to shine in a Liverpool shirt came in their 2-1 League Cup victory over Swansea City on Tuesday and Rodgers hailed his impact ahead of his side's trip to 14th-placed Newcastle.
"Daniel Sturridge being out has been a big miss for us, but sometimes it can give opportunities to others," Rodgers told a news conference on Thursday.
"The other night, Fabio Borini was outstanding. He made a perfect cross at the end and worked hard and was lively all night."
With the misfiring Mario Balotelli finally breaking his six-week scoring drought against Swansea, Rodgers was forced to defend Liverpool's other faltering forwards.
Serbia winger Lazar Markovic has struggled at Anfield since signing from Benfica for 20 million pounds ($32.02 million), while Rickie Lambert has looked lost in front of goal after joining from Southampton.
"We brought young Markovic in for the longer term," Rodgers said. "We have confidence he will become a big player for us.
"Rickie Lambert hasn't had many opportunities. Goals will come. He has worked tirelessly every day."
Seventh-placed Liverpool, who have 14 points from nine matches, travel to a Newcastle side buoyed by their League Cup victory over Manchester City on Wednesday and Rodgers praised the club for sticking by underfire boss Alan Pardew.
"It's a credit to the directors at Newcastle (that they've stood by him). I know Alan is a top class man, but there's pressure if you're not winning games.
"He's a very experienced and a very talented manager, is Alan Pardew.
"An outstanding manager. So his team will get results and we will know that will be a difficult game for us."