There exists a practical need for incentivizing content providers to cache contents at distributed network edges closer to users. However, this is a particularly challenging problem due to system environments that are uncertain, content placements that couple adjacent time slots, and economic properties that are desired but hard to ensure. In this paper, we present our design of an auction-based incentive mechanism for online edge caching. We formulate the long-term social cost minimization problem as a nonlinear mixed-integer program that addresses bid selections, user request dispatching, content placements, and payment determination in repetitive auctions. To solve this problem online, we devise a greedy approximation algorithm for solving each auction individually, and a lazy-replacement-based online algorithm that ties the series of auctions over time while dynamically pursuing the balance between downloading contents to new cache locations and keeping them at existing locations. We formally prove the approximation ratio for each single auction, the competitive ratio for the long-term social cost, as well as the truthfulness, the individual rationality, and the computational efficiency of our approach. Evaluations with real-world data have also validated and confirmed the practical superiority of our approach over multiple alternative algorithms.