It was only his third strike for the club he joined for £16m in the summer of 2011 — and his first since May 8.
But with Young Boys beating Anzhi the goal was the difference between the Reds reaching the knockout stages for the ninth time in 11 attempts and crashing out.
“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, who has been a fringe player for most of the season.
“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 trouble this season has been a reliance on Luis Suarez, playing his first European away match for the Reds, to provide the goals.
Manager Brendan Rodgers has called for others to contribute and in recent weeks Daniel Agger, Jonjo Shelvey, Jose Enrique and even Joe Cole have chipped in.
By hitting the target in the Stadio Fruili, Henderson became the 15th different player to score this season, three short of the club record.
It was not all plain sailing, however, as 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.
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 Daniele Padelli.
In truth, 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.
Henderson must have thought he had doubled his tally in the 65th minute when he perfectly timed his side-foot volley from Jose Enrique’s cross but Padelli tipped it over.
Four minutes later the goalkeeper produced another stunning stop to deny Suarez from close range.
However, just when Udinese started to step up the pressure for an equaliser their chances were diminished by a red card for Giovanni Pasquale. Having already been booked for clattering into Suso, he was shown a second yellow 11 minutes from time for tangling with Sterling.
Substitute Antonio di Natale almost produced the cruellest of endings when he fired over with the last kick but Liverpool survived.
UDINESE: Padelli, Faraoni, Heurtaux, Danilo, Pasquale, Pereyra, Agyemang-Badu, Pinzi (Benatia 46), Armero, Fabbrini, Ranegie (Di Natale 85), Benatia (Reinthaler 65).
LIVERPOOL: Reina, Johnson, Skrtel, Carragher, Jose Enrique, Henderson, Allen, Sahin (Shelvey 12), Fernandez Saez (Sterling 71), Suarez, Downing.
Referee: Duarte Gomes (Portugal).