The opening set proved to be a case of missed opportunities for the young Brit, who had never previously reached this stage at SW19.
She served for the set while leading 5-4 and then spurned a 5-2 advantage in the tie-breaker as Kanepi recovered to win through 8-6.
The Estonian claimed the initiative from then on, although it took her until the 11th game of the second set to claim a break of Robson's serve.
She then went on to earn four match points, but on each occasion Robson held firm. However, at the fifth time of asking, the 28-year-old made a cross-court stroke count to seal her place in the last eight.