Henderson's first European goal put the Reds top of Group A but Udinese, who had Giovanni Pasquale sent off 10 minutes from the end, almost snatched the draw which would have sent Liverpool crashing out with the last kick of the game when captain Antonio di Natale blasted over.
Young Boys' victory over Anzhi meant Liverpool had to win to progress and Henderson's first-half strike meant they finished group winners. Rodgers of the midfielder: "He is a great lad and works hard and has come in over the last month and made a good contribution to the team."
The Reds boss added: "It was a very good finish and he was unfortunate not to get a second goal as the keeper 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."
Had Di Natale scored the inquest would have begun on where it had gone wrong for Rodgers - undoubtedly conceding a late equaliser to Young Boys a fortnight ago - but the miss kept the European bandwagon rolling.
"It is that moment when there is literally 20 seconds left and we've given away the ball. It was very close," admitted the Northern Irishman. "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.
"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.
"We can rest well, focus on the league and look forward to next February."
Udinese coach Francesco Guidolin, whose side had already been eliminated before kick-off, was pleased with the performance.
"I want to thank my team because they have gone out of the competition with dignity and their heads held high," he said. "They showed they could give Liverpool a game."
- Sports & Recreation