Uruguay, who have won the Copa America a record 15 times, and Peru played out a goalless draw in regulation time at the Arena Fonte Nova on Saturday.
Peru eventually triumphed 5-4 on penalties to advance to semi-final stage where they will face winners of the last two editions, Chile, with Suarez the only man who failed to convert his spot kick.
Suarez left the pitch in tears and Neymar, who spent three years with the Uruguayan at Barcelona, took to social media to calm down his former teammate.
"Get up," the injured Brazil forward wrote on his Instagram story, alongside a video of Suarez crying.
"You are very great, brother. I love you friend."
Neymar and Suarez formed a deadly attacking trident during their time together at Barcelona alongside Lionel Messi, winning the UEFA Champions League in 2015.
Neymar missed the Copa America on home soil after suffering an ankle injury on the eve of the tournament but he has still dominated the headlines with reports linking him with a sensational return to the Camp Nou.
Plenty of neutrals will be feeling less sympathetic with Suarez though as the former Liverpool forward has courted controversy in the past with high profile biting incidents and dramatically denying Ghana a place in the semi-finals of 2010 FIFA World Cup when he stopped a certain goal with his hands in extra-time stoppage time.