Liverpool will now play in the Champions League group stage for the first time since 2014, much to Henderson's delight.
"It has been a brilliant night, the atmosphere was brilliant all night and we were brilliant at the start and set the tempo," Henderson said to BT Sport.
"We let in a couple of goals, so we have still got things to work on, but overall we're delighted to get back into the Champions League - where Liverpool belong.
"Getting back into the Champions League was our aim and we've done it. Now when we get into the Champions League itself we have got to do ourselves justice and win some games there."
Liverpool will be placed in pot three in Thursday's group stage draw, raising the prospect of Klopp's team potentially facing the likes of Champions League titleholders Real Madrid, Bayern Munich, Barcelona and Paris Saint-Germain.
But Henderson, who set up Roberto Firmino for a tap-in to secure qualification, insisted five-time winners Liverpool are not fussed about their opponents.
"I'm delighted for the lads and the fans," Henderson said. "We're in there and we've proved we can play against big teams. We're happy to play against whoever we're drawn against."