The Mumbai team was trailing by 8 games after losing the legends singles and mixed doubles but Cornet kept her side's hopes alive with a hard-fought 6-3 win over the Serbian.
Delhi still enjoy a five-point lead over the visitors and need to do extremely well in the last two matches.
Juan Carlos Ferrero yet again provided a great start to Delhi dreams with a convincing 6-2 win over Sergi Bruguera.
He put his side ahead with his tremendous ground strokes, still looking sharp after two years of his retirement.
If the 2003 French Open winner Ferrero dominated with his fierce ground strokes, Bruguera played some awe-inspiring drop shots.
Bruguera, the two-time Roland Garros champion, though was no match to Ferrero as unforced errors flew from the racquet of the 43-year-old unabatedly.
The 34-year-old Ferrero needed only 22 minutes to dismantle his senior compatriot.
With the same scoreline, Kevin Anderson and Jelena Jankovic outplayed the Mumbai pair of Alize Cornet and Tommy Robredo to double Delhi's lead.
However, the Serb found the going tough as Cornet came out all guns blazing, determined to bring her side back in the hunt. The world number 19 Frenchwoman, who has beaten Serena Williams thrice this season, broke Jelena in the fourth game and held the next for a healthy 4-1 lead.
It was a chilly evening with temperature dropping fast but Cornet's court coverage with swift movements and powerful strokes hotted up the tie. She was serving for the set but dropped serve with double-handed backhand.