The midfielder fired home his third Liverpool goal on 23 minutes to put the Reds top of Group A and into the last 32.
"He is a great lad and works hard and has come in over the last month and made a good contribution to the team," said Rodgers.
"It was a very good finish and he was unfortunate not to get a second goal as the goalkeeper made a wonderful save.
"He has that quality to arrive in the box and he has a finish in him so I am delighted for him and his hard work is paying off."
Udinese's hopes of an equaliser looked bleak when Giovanni Pasquale was shown a second yellow card with 10 minutes remaining.
However, the hosts almost snatched a draw that would have eliminated Liverpool with the last kick of the game, captain Antonio Di Natale blasting over the bar.
"Hopefully it is a sign we have a little bit of luck going our way because maybe a couple of months back that would have gone in and we would be out of the competition," said Rodgers.
"There was no margin for error tonight. I felt we had to win the game because we were in a difficult group against good teams.
"Over the course of all the games, how we have played and progressed through the tournament I think we are deserved winners of the group."
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