Jurgen Klopp ended a seven-year barren spell for the Reds in 2018-19 as he guided them to European glory.
A UEFA Super Cup triumph has already been added to the roll of honour this season, having suffered a penalty shootout defeat to Manchester City in the Community Shield.
"It was nowhere near easy. I think it’s going to get harder and harder.
"We know what it’s going to take, there are going to be moments that test us and maybe new tests this year that we didn’t face last year.
"We had to dig in, the boys dug in all last year through difficult moments, where you’re expected to win and still go out and win. We know there’s going to be tests and we know that’s what’s required."
Oxlade-Chamberlain, who made his return to action towards the end of last season following a lengthy spell on the sidelines with a serious knee injury, believes all of the pieces are coming together at Anfield for sustained success to be enjoyed.
The 26-year-old midfielder added: "Mentally, this group of players is top drawer. The manager is one of the most mentally persistent managers out there, for sure, he keeps us going.
"We know what the demands are on each other. And I think that has grown in the team, there are more and more leaders coming out of this side – not just Hendo, Millie, Virg. Everyone is starting to take responsibility for what they need to do on and off the ball.
"It’s a team that’s really going in the right direction. But it’s early in the season; we’ve got to keep our feet on the ground, keep plugging away and keep churning out results."
Liverpool have taken maximum points from their two Premier League games in 2019-20 and will be looking to make it three wins from three when they play host to an Arsenal side on Saturday that has also been faultless so far.