Yerry Mina's header put Everton back in front on the stroke of halftime and James rounded off an excellent individual display with two second half goals.
In the women's 200m sprint, Britain's Awuah, who finished fourth at the Doha Diamond meet last week, clocked 23:05 for her new personal best to beat Hemida Bassant of Egypt, who dipped 23.13 in second place. Zimbabwe's Roda Njobvu was third in 23:18.
Dortmund pressed forward but had to wait until the 31st minute before the good build-up work and pass into the box from Giovanni Reyna allowed Haaland to break the deadlock with a hammer from close range.
The Spaniard, who is seeking his 13th title at Roland Garros, defeated Sinner 7-6 (7-4), 6-4, 6-1 to set up a rematch with Diego Schwartzman on Friday.
The win, which came as the 35th from 36 matches this season, also ties him with Rafael Nadal for a record 14th Roland Garros quarterfinals appearance.
Earlier in the women's race, the world record holder Brigid Kosgei defended her title in 2:18:58.
Han Ziqi contributed a game-high 23 points, but her solo effort could not save Tianjin from a big loss.
Gaston is to challenge third seed Dominic Thiem in the fourth round. Earlier in the day, the Austrian had claimed another straight-sets victory over 28th seed Casper Ruud, beating the Norwegian 6-4, 6-3, 6-1.
1: Barca still has work to do
"If every player can contribute, it will be an easier game for us," she noted.
The 31-year-old, who is to face No. 7 seed Petra Kvitova in the next match, also became the first Chinese singles player since Li Na in 2012 to reach the fourth round at Roland-Garros.