It was not a comfortable journey and it took longer than Brendan Rodgers thought necessary, but a place in the last 32 of the Europa League belongs to Liverpool. Qualification came courtesy of Jordan Henderson's first-half goal plus Antonio Di Natale's last-gasp miss on a night when victory and nerve were essential against Udinese for a place in the knockout phase.

Liverpool finished top of Group A with Anzhi Makhachkala falling to defeat Young Boys in Berne but would have gone out had Di Natale denied them a deserved win in Italy. Instead, the Italy international blazed over following a slick one-two on the edge of the area and could only offer a wry smile, and receive an embrace from a hugely grateful José Reina, as Liverpool advanced on the finest of margins.

"My heart was in my mouth at the end," said Rodgers. "With 20 seconds left we give the ball away and you know Di Natale has the quality to hurt you. Hopefully it is a sign we have that little bit of luck now, because in that one moment we could have been out of the competition, even though we had dominated so much. Over the course of the group we have deserved it, and the fact we are through is great credit to the players."

And to Rodgers, who had to play his hand carefully throughout the group stage due to the lack of depth in the Liverpool squad but successfully navigated the first European test of his managerial career.

There should have been no need for the stoppage-time palpitations, however. By half-time, as Luis Suárez left the pitch in an argument with the Udinese goalkeeper Daniele Padelli, Liverpool were in complete control and the only irritation for Rodgers was the margin of their lead. The ascendancy had been hard-earned, with already-eliminated Udinese far from obliging hosts.

The Udinese coach, Francesco Guidolin, rested several regulars for his dead-rubber, including Di Natale and the first-choicegoalkeeper Zeljko Brkic, while the sparsely-filled Stadio Friuli showed the locals were not tempted to brave a freezing night to offer support. Yet Udinese started brightly and were intent on making their presence felt as the striker Mathias Ranegie collided with Martin Skrtel in the opening minute and caught Nuri Sahin in the face with a flailing arm. Skrtel was able to continue after lengthy treatment but not so Sahin, whose rare start lasted only 12 minutes before he exited with a bloody nose.

Jonjo Shelvey was introduced from the bench, where he started in readiness to lead a Suárez-less attack at West Ham on Sunday, and was soon in a heap following a dangerous challenge from Diego Fabbrini. The Udinese forward was only booked for what appeared a two-footed tackle.

Left-wing crosses provided Udinese's most consistent threat on the Liverpool goal and Ranegie glanced a header wide from Pablo Armero's delivery before é Reina, having squandered possession with a poor clearance, redeemed himself with a good save from Roberto Pereyra's audacious chip. But once Liverpool edged in front midway through the first half the fight faded from Udinese.

Suárez was integral to the breakthrough on his first European away appearance for Liverpool. His free-kick struck the wall and was eventually deflected behind for a corner taken by Stewart Downing. Suárez peeled away from his marker to win the header, the ball fell for Suso at the back post and he had the presence of mind to set up Henderson for a simple finish into the far corner.

Rodgers said: "That goal will do Jordan the world of good. He's a great lad and he's worked very, very hard. He's come into the side in the last few months and made a great contribution to the team. It was a great finish and it will give him confidence."

Suso and Suárez squandered good chances to extend the lead and, but for an often poor choice of final ball, Liverpool would have been comfortable long before the introduction of Di Natale injected some quality into an Udinese team reduced to 10 men when Giovanni Pasquale received a second booking 10 minutes from time. They were also frustrated by Padelli, who made one appearance for Liverpool in 2007, who twice thwarted Suárez with fine saves and denied Henderson a second from close range.

"There was no margin for error," the Liverpool manager added. "We knew we had to win because we were in a difficult group with a Champions League team and one of the richest teams in the world. The desire of the team to win the game was very strong and that is good for the future."