It was the biggest win of Yuki's career when the formidable Italian, who struggled due to an injured left leg, conceded the match after trailing 1-6 5-5.
Fognini had earlier pulled out of the doubles event for which he was supposed to partner India's veteran star Leander Paes. They duo were seeded second.
Yuki, ranked 195, is only the third Indian after Somdev Devvarman and Leander Paes to make the singles quarterfinals of this event in Chennai.
Despite his restricted movements, Fognini was good enough to trouble Yuki with his solid ground strokes, specifically his power-packed forehand.
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A proper contest would have served as a good tennis lesson for the Indian youngster. It wll be first ever appearance for Yuki in an ATP tour tournament quarterfinals tomorrow and it will help him improve his ranking.
"It's great to be in the quarterfinals of an ATP tour event before the home crowd. You don't get to play matches against world number 16 everyday. I am looking forward to next match tomorrow," Yuki said.
After breaking Fognini in the fourth game, Yuki raced to a 3-1 lead but had to battle out two breakpoints in the next game before comfortably serving out the set in the seventh game.
In the second set too, Yuki succeeded in getting the first break in the fifth game but was broken back. It was on serve at 5-5 when Fognini decided against continuing.
His next opponent would be either Canadian world number 32 Vasek Pospisil or Taipei's world number 66 Yen-Hsun Lu.