We consider load balancing problem in a cache network consisting of storage-enabled servers forming a distributed content delivery scenario. Previously proposed load balancing solutions cannot perfectly balance out requests among servers, which is a critical issue in practical networks. Therefore, in this paper, we investigate a coded cache content placement where coded chunks of original files are stored in servers based on the files popularity distribution. In our scheme, upon each request arrival at the delivery phase, by dispatching enough coded chunks to the request origin from the nearest servers, the requested file can be decoded. Here, we show that if $n$ requests arrive randomly at $n$ servers, the proposed scheme results in the maximum load of $O(1)$ in the network. This result is shown to be valid under various assumptions for the underlying network topology. Our results should be compared to the maximum load of two baseline schemes, namely, nearest replica and power of two choices strategies, which are $Θ(\log n)$ and $Θ(\log \log n)$, respectively. This finding shows that using coding, results in a considerable load balancing performance improvement, without compromising communications cost performance. This is confirmed by performing extensive simulation results, in non-asymptotic regimes as well.