On the first day of the third leg in Delhi's Indira Gandhi Indoor Stadium, Aces made the most of home advantage to win through on a Super Shootout and secure a result that keeps them top of the pile.
After four tight sets early on, Mavericks enjoyed a four-game cushion following Jo-Wilfried Tsonga's 6-4 victory over Monfils in the men's singles.
However, that deficit was wiped out by Aces' world number five Ana Ivanovic, who brushed aside Kirsten Flipkens 6-2.
That took the encounter to a deciding finale, where Monfils was able to prevail against his countryman Tsonga.
In the day's early match, UAE Royals gained a measure of revenge against Singapore Slammers with a comprehensive 29-16 victory.
Slammers got their first win of the competition against Royals in Singapore, but were comfortably beaten on Saturday, having got off to a strong start courtesy of Marin Cilic.
The Croatian was 3-0 ahead early on against Lleyton Hewitt before wrapping up a 6-3 win.
Cilic teamed up with Nenad Zimonjic to clinch the men's doubles against Hewitt and Nick Kyrgios before Goran Ivanisevic brushed aside Pat Rafter 6-1 to give Royals an 18-6 advantage.
Zimonjic and Kristina Mladenovic edged Royals closer to a dominant win courtesy of a mixed doubles success, with Daniela Hantuchova's single's win against Caroline Wozniacki a mere consolation to leave Slammers bottom.
Speaking afterwards, Cilic explained: "I am glad to have played well in the opening match.
"Our coach [John-Laffnie de Jager] asked me to play slow, take it easy, and that's what I did. The match was exciting with two Power Points and I had fun.
"Now that I have played six matches I am much more confident and it feels good."