The Blues defeated champions Liverpool 3-1 on Sunday, though Elliott says the visitors ended the game the better team.
Elliott told Liverpoolfc.com: "I think the first half was very hard. It was hard to find the space, to create the chances. I think we had a few counter-attacks and they just fizzled out.
"But we reflected on that at half-time and I think in the second half there was a massive change. I felt like we all clicked into gear and I felt myself personally there were a few things that were going for me, a few half-chances you could say.
"But I think as a whole in the second half we were a lot better. Obviously the skipper got the goal to make it 2-1 but I think it was just a little bit too late.
"But it's something we can reflect on throughout this week and it's another big test again on Sunday against Arsenal to go and make it right. Let's hope we can do that."
"We're now just very relaxed"
Elliott also explained: "I think every human being in our shoes now is very relaxed, very composed and all that stress and hard work throughout the season to try to make it an achievement, we've now done it.
"I'd say it's a relief but at the same time we still have three more games and we need to make sure that we put 100 per cent into it.
"We need to go out and put in top performances not only for us but to show why we are champions. We need to do that again next week."