The world No1 has been searching for top form all season and finally found something close to it on a cool, grey day at Roland Garros.
The world No 1 battled back to win a marathon first set lasting one hour and 25 minutes before triumphing 7-6 (8) 7-5 6-0 on Philippe Chatrier.
It looked a nightmare draw against a player falsely low in the rankings as he continues to work his way back up after wrist problems that nearly ended his career.
"Obviously from his reaction at the end of the first set, he was pretty disappointed", Murray said.
Cornet reached the last 16 for just the second time with a 6-2, 6-1 win over Polish ninth seed Agnieszka Radwanska who lost all her seven service games. To be playing him this early on in a Slam is not easy, but it can be a very positive thing.
That match was one of six still to be completed when rain halted play. That's why he's the No. 1 of the world at the moment, and I know how important is this tournament for him.
"But I'm not thinking about winning [the tournament] after each round. I'm under the impression that I have just come out of bed". I was going to try and just get through the first round and hopefully feel better each day. That's been the case so far.
"If I say something, you guys ask me what I was saying; if I say nothing, you ask me why I don't say anything". I knew that if I stayed behind the baseline, he can only do so much. "So I like to play chess in my free time".
Andy Murray had what appeared to be a hard third round encounter with Juan Martin del Potro but the Scotsman delivered on the big stage. The players split the first two sets.
At 2-3 Del Potro summoned on the trainer and was given two pills, presumably to ease any soreness in the groin area, although his movement did not look hindered.
The 30-year-old said: "I believe in myself, so even when things aren't going well, I believe I can turn it around". "I have been playing great points during the whole first set". But the Tower of Tandil was not quite done breaking Murray as served for the second set at 5-4.
In the women's event, France have 3 women in the last 16 for the first time in 23 years.
With some exquisite returning and acutely angled groundstrokes, Murray raced through the final set to set up a fourth-round clash with either American 21st seed John Isner or Russian Karen Khachanov.
She played her typically safe style for much of the match, winding up with 15 fewer unforced errors than No. 8 Kuznetsova.
Murray is a three-time grand slam title victor and was the runner-up at Roland Garros a year ago.
They each had a significant win previous year with Murray taking gold at the Rio Olympics final and Del Potro edging a five-set thriller when Argentina beat Britain in the Davis Cup semifinals.
He said: "I already congratulated him with past results".
· Get Tennis Channel Plus for Roland Garros to watch 200 live matches from up to five courts, starting with the Qualifying Rounds. He was a real number one. That's why he has all the shots to play but also a great mentally. This is one of the most important tournaments of the year.
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