In a match dominated by serve, Kanepi drew on her greater experience to prevail, 76(6) 75, and advance to her fifth Grand Slam quarterfinal. However, Robson, who cut a disconsolate figure as she sloped off court at the end, was not without her chances.
After knocking on the door on a couple of occasions, it was Robson that struck first, dishing out another unreturnable forehand to break for a 5-4 lead in the opening set. Yet she was unable to capitalize on this breakthrough as Kanepi delivered an instant riposte - a vicious crosscourt backhand winner set up a break point, which she duly converted when Robson sent a forehand long.
The Brit came even closer to the set in the ensuing tie-break, a flurry of winners helping her to 3-0 and 5-2 leads. Once again it was not enough; Kanepi rattled off six of the next seven points to take the opener.
With Kanepi now growing in confidence, the Brit was forced to batten down the hatches, three times facing break point at the start of the second and three times being bailed out with swinging aces out wide.
There was no reprieve in the 11th game, though, as the Estonian broke to love.
Robson came back from the brink to defeat Marina Erakovic in the previous round and, roared on by a partisan crowd, she threatened another comeback, saving four match points. There was to be no escape on number five as Kanepi wrapped up victory with one final clubbing forehand winner.
"I didn't think too much about the fans, I just tried to focus on myself and my opponent," Kanepi said. "In any case I think the crowd was very intelligent - they didn't clap when I made a bad shot or a double fault, so they were nice!
"I think my serve and attacking game are working well for me, which is important on grass, and I feel comfortable on the return too. If I'm playing well and I'm on a roll, I enjoy the surface but overall I have to say it's my third favorite after clay and hardcourts.
"It was a very tough match and I'm happy to get through it, but I will need to be even better for the next round."
"Sabine had a great win and she is obviously playing very well," Kanepi said. "She has such a big game so I guess I'll have to hit it very hard too!"