The Boks won the opening encounter 42-24, but they were not playing a full-strength Italian team and there was clear frustration from head coach Rassie Erasmus and captain Jesse Kriel following the result.
Peyper says the Boks are regrouping with intent and accountability as they look to raise their level for this weekend's second Test.
"The coaches always sit down and look at whether we planned well enough and if our execution was good enough, and we did that," the former international referee told reporters on Monday. "We are facing a passionate and well-coached Italian team, and we need to be up for the game and work hard this week to deliver on our standards.
"We are a proud team, and that's why we were frustrated. There were too many individual errors, and if those stack up, it mounts up through the game."
Springbok prop Thomas du Toit also fronted up to the media, and said the players had acknowledged they did not play to the standards they had set for themselves.
"I wouldn't say that it was a wake-up call," he said. "Obviously, we could have been much better last week, and there were certain aspects of the game in which we had to look ourselves in the eye and say we were not good enough."
Du Toit said the Boks had to respect the way Italy fronted up at Loftus.
"They are a young team with a lot of energy, and they are well-coached. They don't mess around in their own half, and from our perspective, we need to ensure that we don't allow them to get into our 22 and make their presence felt there.
"We respect them and their patriotism for their country, as that's something that resonates with us. There is a good alignment in their squad, and they are gelling well as a unit, so there are definitely areas we need to improve on this week, while we also have to build on the areas of our game which went fairly well."
Meanwhile, this Saturday's clash at the Nelson Mandela Bay Stadium could see Bok veteran Willie le Roux become only the eighth Springbok to earn his 100th Test cap.
"Willie is an unbelievable player, and if he gets to play his 100th Test, we will be very proud of him," said Du Toit. "It speaks volumes about the player that he is, and it's important to have him in the team. One thing's for sure, if he does play, we will try to make it as special as we can for him."
The Springbok team to face Italy will be named on Tuesday afternoon.