Defeated 6-7 (6-8) 5-7, the British number two was the superior player for the majority of the first set but failed to serve out.
Jo Durie remains the last British woman to reach the quarter-finals at the All England Club, in 1984.
Kanepi, ranked 15 last year, held and saved a break point in two taut opening service games whereas Robson was exceptional on serve.
The Putney girl finally capitalised on her opponent's unsteady serve in the ninth game, breaking Kanepi ahead of serving for the opening set.
Kanepi appeared overawed by the partisan Court No. 1 but bullishly and belatedly asserted herself. Robson's serve, potent yet mercurial, tightened as Kanepi upped the aggression to commandingly break back and hold, as the momentum swung the Estonian's way and Robson had to settle for a tie-break.
5-2 up, Robson's loopy ground strokes intervened and she served an abject double fault to gift Kanepi the chance to serve for the set. Although the teenager saved set point it was merely a stay of execution, and just like against Marina Erakovic on Saturday Robson lost the opener.
Robson saved a break in the opening game of the second following another double fault, as the advantage of opening serve failed to register.
The Melbourne-born Brit saved two break points with aces at 3-3 for a crucial hold, followed by a confidence-boosting service game for a 5-4 lead.
Kanepi, who claimed she would "enjoy" the crowd siding against her, broke for 6-5 as Robson's infamously languid movement began to restrict her.
The world number 46 eventually served out after Robson gutsily saved four match points to end her week in the SW19 spotlight.