The event at Abu Dhabi Golf Resort gives European players valuable matchplay action ahead of the Ryder Cup in New York this year.
Britain and Ireland led 3½-1½ overnight following Friday's fourballs and won 7½ of 10 points during the foursomes sessions on Saturday.
Britain and Ireland now need just two points in Sunday's singles to clinch the trophy after a performance which has so far delighted captain Justin Rose.
"The leaderboard got more and more and more in our favour as the rounds went on and on and on," Rose said.
"Every match that got tight, we either got back to halve or turned them around to win. Today was the big day, 10 points up for grabs. They have done an unbelievable job."
Englishman Tommy Fleetwood stood out with victories in both sessions, teaming with countryman Rose in a 2 up victory over Continental Europe captain Francesco Molinari and Julien Guerrier of France.
Fleetwood then collected three points alongside Jordan Smith in a 3&2 win against the same pair in the afternoon.
"We were just very relentless tee to green today," Fleetwood said. "We played very well on what I thought was a pretty difficult afternoon."
Northern Irishman Tom McKibbin matched Fleetwood's effort. He partnered Aaron Rai to a 4&3 victory over Rasmus Hojgaard and Rasmus Neergaard-Petersen, and joined Matthew Jordan for a 2 up win against the other Danish pair, Thorbjorn Olesen and Niklas Norgaard.
Frenchmen Romain Langasque and Matthieu Pavon fought well to win both of their matches for Continental Europe, after also winning their fourball match on Friday.
"We were great today," Langasque said. "Tomorrow is going to be a tough day for us as a team because we are way behind."
Continental Europe must sweep Sunday's singles to retain the trophy won in 2023, when it was called the Hero Cup.
The event was formerly called The Seve Trophy and was played eight times from 2000-13, with Britain and Ireland winning six times and Continental Europe winning twice.
Six of the 20 players and captains from the 2023 edition wound up playing the Ryder Cup that year, when Europe won to reclaim the cup.
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