The Reds triumphed 1-0 in Italy to secure top spot in Group A thanks to Jordan Henderson’s first half strike, while Young Boys defeated Anzhi 3-1 in Switzerland.
And Rodgers believes his side fully deserve to be the group winners.
“"I thought we were fantastic tonight,” he told reporters.
"We had two objectives coming into this stage of the tournament. Firstly, tonight was our 10th game in the tournament and qualification was important. Secondly, we wanted to give some experience to our younger players.
"When you're 17, 18, 19 and 20 years of age like some of them are, this is a big learning curve.
"Our challenge is to make them better players and educate them in the game. They have done well tonight.
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"But the mentality of the team was very strong and our desire and belief was very strong. We won and that is good for our future.
"I'm delighted for the players and supporters, over 1,000 of whom travelled over here and that was important. They deserve all the credit along with the players. We can now look forward to the next round in February."
Henderson was his team's hero on a night where Liverpool needed a result, and Rodgers praised the 22-year-old’s display, hailing the effort he has shown in improving his form in recent times.
"His industry tonight was superb,” Rodgers added.
"He is a great lad. He works very, very hard and he's come in over the last month or so and made a good contribution to the team.
"He was unfortunate not to get a second goal - the 'keeper made a wonderful save - but he's got that quality to arrive in the box and he's got a finish in him.
"I'm delighted with him and his hard work is paying off."
Liverpool's win and the result in Switzerland meant they topped Group A, thanks to a better head-to-head record against Young Boys and Anzhi.
And Rodgers is glad the Reds have continued their momentum in the competition.
"It was always going to be a very tight group and you saw how close it finished," he continued.
"I said to the players I thought we could finish top of the group with a good result tonight.
"I'm absolutely delighted for them because they have been superb throughout the campaign, and their Premier League form is improving all the time.
"It's really good for our momentum to qualify."