Williams looked in ominous form as she won the first in just 18 minutes, making just two unforced errors.
But Kanepi hit back to break immediately in the second set and got an insurance break to lead 5-2.
Williams pegged Kanepi back but the Estonian came through 6-4.
However, the 23-time Grand Slam winner showed her quality on serve in the final set winning 16 out of 17 first serve points to win the match 6-0 4-6 6-3 and she will face Karolina Pliskova next after she beat Ash Barty.
Serena Williams vs Kaia Kanepi AS IT HAPPENED
Williams* 6-0 4-6 6-3 Kanepi*
Ace. Kanepi forehand into the net. Then a backhand into the net. 40-0. Game. Set. Match.
A fist pump by Williams as she comes through her toughest match of the US Open so far.
Williams 6-0 4-6 5-3 Kanepi*
Kanepi ace to start the game. Williams then goes long with a forehand return. Kanepi cross court forehand winner.
Williams returns long. She will now serve for the match.
Williams* 6-0 4-6 5-2 Kanepi
More good serving gets Williams to 40-15 and yet another ace seals the game.
The best serve in the women’s game is making all the difference.
Williams 6-0 4-6 4-2 Kanepi*
Game Kanepi as Williams goes for too much. She wants to get this match done as soon as possible.
Williams* 6-0 4-6 4-1 Kanepi
Ace no 15 for Williams as she blasts her way through Kanepi. Two games away from the quarter-finals.
Williams 6-0 4-6 3-1 Kanepi*
Kanepi stops the rut and holds to love as Williams’ level just drops slightly.
Williams* 6-0 4-6 3-0 Kanepi
A 12th ace but then a fourth double fault makes it 30-all.
Williams forehand wide gives Kanepi a break point. Wild Kanepi backhand sends it to deuce.
Ace no 13. And Williams clings on. Both women hitting huge groundstrokes.
Williams 6-0 4-6 2-0 Kanepi*
15-40. As Williams forces Kanepi into errors and she breaks through to lead.
Williams* 6-0 4-6 1-0 Kanepi
Forehand winner by Williams to move to 40-30. Another huge Kanepi forehand winner keeps the pressure on.
But Williams smacks a backhand cross court winner to take the game and she lets out a massive roar.
She wants this badly.
Second Set: Williams 6-0 4-6 Kanepi*
Another 0-30 situation on serve. Williams puts a forehand into the net 15-30. Big backhand by Williams forces an error from Kanepi.
Two break points. The first is saved and then the second with a wide serve.
Set point. Williams keeps herself in the game with a smash winner. Second set point. Williams backhand down the line winner.
Third set point as Kanepi hits a forehand winner. Williams goes wide with a return and it’s all square on Arthur Ashe.
Williams* 6-0 4-5 Kanepi
This time it’s Kanepi’s turn to go 0-30. But a Williams ace brings it back to 30-30.
Wow. Blistering forehand winner from Kanepi sends it to deuce.
But Williams and puts the pressure back on Kanepi to win the set on her serve.
Williams 6-0 3-5 Kanepi*
0-30 as Williams goes on the attack and rifles two backhand winners.
Then a forehand winner gets one of the breaks back for Williams.
Williams* 6-0 2-5 Kanepi
Kanepi has been much improved in the second set and he holds serve again and gets another break point on the Williams serve.
And Kanepi gets an insurance break when Williams hits a double fault. A complete turnaround.
Williams* 6-0 2-3 Kanepi
Big serving from the Estonian gets her through her service game. A love service hold for Williams follows.
Williams 6-0 1-2 Kanepi*
Kanepi consolidates her break of serve to continue her fightback and Williams then keeps hold of her own.
Williams* 6-0 0-1 Kanepi
Another break. But not for Williams. Kanepi finally gets on the board when Williams goes long. Game on?
First Set: Williams 6-0 Kanepi*
18 minutes. That is how long it took for Williams to win the first set. Just two unforced errors. Ruthless performance.
Williams 4-0 Kanepi*
Perfect start maintained by Williams as she holds to 15 and breaks Kanepi again. Ominous.
Williams 2-0 Kanepi*
Immediate break points for Williams and she takes them to move ahead.
21:00: Welcome to LIVE coverage of Serena Williams’ US Open fourth round clash against Kaia Kanepi.