Robson went a break ahead early on, but Kanepi's power from the back of the court proved too much for the 19-year-old and she took the first set on a tie-break.
Robson lost her temper on a couple of occasions and it showed in her play as she succumbed to a 7-6 (8/6) 7-5 defeat, ending her hopes of becoming the first British woman to reach the quarter-finals of Wimbledon in 29 years.
Robson trudged off the court with her head down, but could still be proud of what she has achieved at Wimbledon this year.
She will be inside the world's top 30 for the first time when the rankings are updated next week.
There was no shame in losing to former Wimbledon quarter-finalist Kanepi, but it is clear that there is still work for Robson to do on her game, particularly with her forehand, which she continues to over-hit.
After starting her last match terribly, Robson made a confident start on Monday in front of a noisy crowd. Indeed it was Kanepi who seemed to be letting the occasion get the better of her.
A double fault from the Estonian handed Robson her first chance to break, but Kanepi came up with the goods, firing down a deep serve which her opponent could only return wide.
Robson won a baseline rally and sent down a big ace to win her next two service games to love.
Kanepi regained some confidence briefly, but she then wilted on serve, offering up three break points. Robson wasted the first, over-hitting a forehand, but she took the second opportunity, hitting a bruising forehand return after a weak second serve from her opponent.
It was Robson's turn to feel the nerves as she blazed long to allow Kanepi to break back immediately. The teenager then turned the air blue with a loud curse.
A tie-break followed and Robson claimed first blood, going a mini-break up when Kanepi netted. Robson was 5-2 in front but then hit trouble, losing four straight points, soon allowing Kanepi to break and seal the set.
Robson was on the ropes early in the second set, handing Kanepi a break point in the first game. The British number one responded perfectly though by hitting a 99mph ace before going on to hold.
A huge squeal in frustration from Robson came as she lost the sixth game of the set to love. The tantrum made her lose her focus as Kanepi gained two break points, but Robson answered by sending down a pair of aces.
Robson continued to struggle to cope with Kanepi's power. The Englishwoman changed her tactics in the eighth game, momentarily leaving the baseline.
The key blow came in the 11th game. Serving to move 6-5 ahead, Robson capitulated, offering Kanepi three break points. The Estonian needed just one as Robson found the net again.
That offered Kanepi the chance to serve out for the match. Robson saved two match points to take the game to deuce and Kanepi wasted two more chances to win.
The home crowd sensed a possible comeback, but Kanepi put paid to those hopes with a typically big forehand from the back of the court.
Kanepi said before the match that she was looking forward to playing in an intimidating atmosphere.
But the Estonian was surprised that the crowd did not much pressure on her on Monday.
"I think the crowd wasn't that bad actually," Kanepi told the BBC.
"When she won a point it was a bit louder than normal but they didn't clap when I double-faulted or anything. I think it was really good."
Kanepi is one of the biggest names in Estonian support. When asked how her victory would be received in the Baltic state, the 28-year-old said: "I don't know, maybe with champagne."
The Estonian was pleased with her performance on Monday and she is now confident of progressing beyond the quarter-finals.
"My serve and baseline game were good today [Monday] and I guess I will have to hit hard in my next match, that's always important on grass," Kanepi added.
Robson made her dissatisfaction at failing to make the last eight clear during her post-match press conference.
She said: "I'm really, really disappointed. I thought she played a really solid match. She can hit the ball incredibly hard off the ground, so it was tough for me to stay in the rallies. But I had my chances here and there and I just didn't take them."
After going through what she described as a "crazy" nine days which included messages of support from the biggest names in tennis, showbiz and politics, Robson struggled to deal with her own expectations.
"I was putting a lot of pressure on myself," the 19-year-old said.
"At the end of the first set I had my chances. I served for it and I did in the tie-break as well.
"At that point, I was just trying to will myself to play unbelievable tennis when just making a serve would have been fine.
"But as cliched as it sounds, it's all part of the learning experience. The more I get myself into those kinds of situations, the more I'm going to benefit."
Robson had ended her last three games by saluting the crowd that had come to cheer her on, but today she failed to do so.
"That was because I lost, and I was just trying not to cry," the teenager added.