Cornet, the No.3 seed this week, set the tone for the match with a break in the very first game and by the time Bratchikova began to find her feet half an hour later the first set had come and gone.
The Russian made a better fist of things in the second, but her fate was sealed when she came off second best during a run of three straight breaks midway through the set.
"I'm happy to have played my first match here and also that it was a good one," Cornet said after her 61 63 win. "I'm more than ready for my next match."
Radwanska reached a new career-high ranking of No.40 on Monday, but María-Teresa Torró-Flor, her opening opponent in the Uzbek capital, seemed intent on raining on her parade. After losing a tight first set, Torró-Flor suddenly burst into life in the second, a flurry of winners seeing her race into a 5-1 lead.
However, from here the World No.112 seemed to run out of steam, winning just five points in the next five games as Radwanska pegged her back before eventually triumphing, 64 75.
"I had never played her before and didn't know what to expect," No.2 seed Radwanska said. "She played some unbelievable tennis in the second set to go up 5-1, but I told myself she cannot be playing like this the whole match.
"I knew I needed to put some more balls in court and make her make mistakes, so I kept my patience and played my strokes to come back and take the match."
No.7 seed Panova was given even more of a scare against Jarmila Gajdosova. After splitting the opening two sets, Panova trailed 3-1 in the decider, but capitalized on some jittery serving from her opponent to come back and eventually prevail, 46 62 75.
Sharipova, who is the youngest player in the draw and required a wildcard into the tournament, was particularly pleased with her, 64 75, win, having lost to the same opponent at an ITF event in Kazakhstan earlier this year.
"I had lost to Vlada just a month before and was nervous during the match, but I played well at the end of the match in spite losing a 4-2 lead," Sharipova said. "I'm happy to maintain my points from last year and I can hopefully now go even further."