Sarries will begin the defence of their domestic title against Worcester in the London Double Header at Twickenham on Saturday, 3 September.
Barritt told BBC Radio London: "Last year was probably the most successful season in the club's history.
"But we now have a fresh page to write a new chapter in the Saracens story."
England centre Barritt, 29, played a key role in Saracens' momentous campaign, which culminated in their 28-20 victory over Exeter Chiefs in the Premiership final at Twickenham on 28 May.
Two weeks earlier, they were crowned European champions after beating Racing 92 in Lyon, becoming the first team in the competition's history to win all nine games in the tournament on their way to picking up the trophy.
However, Barritt believes they could face an even tougher challenge next time around.
"This Premiership shows you that anyone can beat anyone on their day. You can't be fooled into any false sense of safety that it's just going to happen," he said.
"The Premiership is the most competitive domestic league in the world. There's quality throughout and every team brings equal and different challenges."
Barritt also said the "continuity of the squad" - of which eight were part of England's Six Nations-winning squad earlier this year - has been "a huge key" in their success.
"We've been a very settled squad who understand each other, who've played together for a certain amount of time, and that's allowed those bonds to grow," Barritt added.
"You can see by the intensity of training and the focus in the players' minds that the guys are very eager to make it another successful year."
Meanwhile, Saracens are learning from the success of Australian Football League club Hawthorn Hawks as they seek to deliver another trophy-laden season.
Hawthorn have won the last three AFL Championships and sit top of the table as they bid for an historic fourth in a row.
"We've looked specifically at the Hawthorn Hawks in the AFL, who are [aiming for] a fourth championship in a row," Barritt told BBC Radio 5 live.
"It's about the relentlessness of the organisation. It's about an organisation being hungry and being as driven as they have ever been, as if it's their first championship.
"We are back-to-back Premiership champions and (now) European champions. We know those things don't come easily, and we know the next season is going to be that much harder because you are going to have a target on your back."
Premiership and European Champions Cup winners Saracens are determined to be even stronger next season, says skipper Brad Barritt.