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Federer and Djokovic will face off on Thursday in a repeat of this year's Wimbledon final to determine who else will progress from Group Bjorn Borg.
The six-time champion was not at his fluent best on Tuesday but ultimately had too much for Berrettini, who won just three games against Novak Djokovic in his opener.
Team Europe's Nadal was also scheduled to take on Team World's Nick Kyrgios in his singles clash.
Rafael Nadal and doubles partner Stefanos Tsitsipas were beaten by Nick Kyrgios and Jack Sock in the final match of the day as Team World cut down Team Europe's lead to 7-5.
Austria's Dominic Thiem, Greek Stefanos Tsitsipas, Russian Karen Khachanov and Spaniard Roberto Bautista Agut were all sent packing before Rafael Nadal restored some semblance of order to his half of the draw.
Pella's win over the Canadian took him to his first major quarterfinal, where he will face Spaniard Roberto Bautista Agut.
The trio, with 14 Wimbledon titles between them, have been in blistering form while players such as Alexander Zverev and Stefanos Tsitsipas -- younger men touted to end their hegemony at the majors -- went out with a whimper.
Gauff, the youngest woman ever to survive the qualifying tournament, played like a seasoned performer as she clinically beat the 39-year-old great 6-4 6-4 to earn a standing ovation from an enchanted Court One crowd.
The 37-year-old is now the oldest man to reach the quarterfinals of a Grand Slam since 1991.
The 20-time Grand Slam winner returned to the Roland Garros for the first time in four years.