Serena Williams has fought off the challenge of giant-killer Kaia Kanepi to claim a rollercoaster win and move into the US Open quarter-finals, a step closer to a record-equalling 24th career grand slam title.

After a shock upset of world No.1 Simona Halep in the first round, the 44th-ranked Kanepi had looked a potential trouble spot for Williams and pushed her to a third set before the American prevailed 6-0 4-6 6-3.

“It wasn’t an easy match at all. She obviously knows how to play, she’s had a lot of big wins,” Williams said.

“We always have tough matches. I was just happy to get through it to be honest.”

Asked about the improvements she has made since losing in the final of Wimbledon to Angelique Kerber, Williams said: “The thing I’ve improved most is my fitness. Coming back from a baby is hard, harder than I thought.”

Next up for Williams is a last-eight showdown with Ashleigh Barty’s conqueror, Czech eighth seed Karolina Pliskova.

Any early tension inside a packed Arthur Ashe Stadium was eased as a ruthless Williams stormed through Kanepi and the opening set in 18 minutes for the loss of a measly six points.

But in a bizarre twist in momentum, fans suddenly found themselves back on the edge of their seats as the Estonian opened the second with a break and broke Williams again to go up 5-2 before holding off a late rally from the 17th seed to level the match.

Having collected her first set off of Williams in five career meetings, however, there would not be a second.

Just as quickly as the momentum had swung to Kanepi in the second set it returned to Williams in the third as normal service resumed with the 36-year-old breaking her opponent at the first opportunity and jumping out to a 3-0 lead before she closed out the match.