IF there is one cliché that rings more true than anything on the women's Tour, it is this: Don't, whatever you do, get Serena Williams angry.

Ekaterina Makarova's win over the American at the Australian Open this year was the biggest of her career, but paradoxically the worst thing she could have done, because Williams will never forget it.

Yesterday, in brutal conditions with temperatures well over 90 degrees Fahrenheit, the Wimbledon and Olympic champion gained revenge for that defeat with a 6-4, 6-0 victory that, despite being tougher than the scoreline would suggest, sent her into the last 16 in convincing fashion.

"At the end of the second set, I started playing a little better and a little more consistent," Williams said.

"Today I knew that I had to play better – she is such a strong opponent and so I knew that I would have to improve on my first two matches. I think I did that."

Trying to win the US Open for the first time since 2008 is probably motivation enough for Williams, but there seemed to be an extra spring in her step, steel in her eye and a determination that she would not suffer again, as she did in Melbourne at the start of the year.

The left-handed Makerova, ranked 42, is a talented, awkward player and with some wicked angles on her serve, she was able to pull the American out of court and take advantage of the open space.

The first four games lasted 20 minutes and were gruelling in the extreme with both players missing chances and saving themselves with some outstanding tennis.

The problem with playing against Williams, though, is that her attacks are so relentless and so suffocating that there is usually an air of inevitability, and having been untouchable on her own serve, she broke in the 10th game to take the set.

In such heat and staring at Williams across the net, it was always going to be a tough task for Makerova to get back into the match and so it proved.

An early break in the second set was enough to finally break her resistance and from then on there was no danger for Williams, who eased to victory and a meeting with another Russian, 14th seed Maria Kirilenko, or Andrea Hlavackova of the Czech Republic.

Second seed Agnieszka Radwanska, who was beaten by Williams in the Wimbledon final in July, had to battle to get past former world No 1 Jelena Jankovic 6-3, 7-5 to take her place in the last 16.

The Pole, who plays Italy's Roberta Vinci next, said it was almost like playing against a mirror, the two baseliners trading baseline blows in similar fashion. "That's why I think we always played long rallies, long games, long points," she said.

Jankovic is not the player she was when she became world No 1 in 2008 and having dropped down to No 27, her confidence is nowhere near where it was four years ago. Radwanska said: "She's a very consistent player,. We played couple of times against each other, and they were always long and tough matches. I was prepared for that."

Vinci gave the Italians something to shout about as the world No 19 upset 13th seed Dominika Cibulkova of Slovakia 6-2, 7-5. Looking ahead, Radwanska said: "She's another tough opponent. I just have to do everything well against her."