The result sent the Reds through to the knockout stages as winners of Group A.
England international Henderson scored the only goal of the game, turning in a pass from Suso, which delighted his manager:
“He is a great lad and works hard and has come in over the last month and made a good contribution to the team,” he told SkySports.
“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.”
Despite being reduced to ten men in the closing stages, Udinese almost drew level with the final kick of the game.
The Italian side’s captain Antonio Di Natale sent a stinging effort just over Pepe Reina’s crossbar and Rodgers was glad to his side benefit from a little luck:
“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.”