"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 of the midfielder. "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."
Liverpool's game plan had to be altered in just the 12th minute when Nuri Sahin was forced off with a bloody nose after getting caught in the face, meaning an early introduction for the Reds' leading European scorer this season Shelvey.
The hosts, despite having nothing to play for, created the early chances with Mathias Ranegie heading wide and Roberto Pereyra forcing goalkeeper Jose Reina to tip over his effort.
Suarez's first shooting chance came from a free-kick 20 yards out but he drilled it into the wall but Henderson took full advantage of his first sight of goal.
A month ago against Anzhi in Moscow the England Under-21 midfielder had been hesitant in front of goal but he had no time to be in the 24th minute.
Stewart Downing's corner dropped to Suarez and although his header only reached Suso the young Spaniard had the presence of mind to lay the ball back for Henderson to drill a shot past Padelli.
The visitors failed to properly test Udinese's second-choice goalkeeper, who made just one appearance in May 2007 for Liverpool while on loan from Sampdoria.
Udinese briefly threatened when Reina had to make a low save from Fabbrini's long-distance shot but Liverpool finished the stronger.
Suso looped over a stooping header from Shelvey's cross and Suarez, having fired over an angled shot, forced Padelli into a fingertip save with a close-range overhead effort from the last kick of the half.
Suarez and Suso both had shooting opportunities early in the second half but failed to hit the target while Ranegie powered a header just over as the hosts pressed for the equalizer.
"It is that moment when there is literally 20 seconds left and we've given away the ball. It was very close," admitted Rodgers. "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."
Having closed out only their fourth victory in 10 matches on Italian soil, the Reds can now put Europe to bed until February and focus on improving their Premier League position.
Information from The Associated Press and Press Association was used in this report.