For edge coloring, the online and the W-streaming models seem somewhat orthogonal: the former needs edges to be assigned colors immediately after insertion, typically without any space restrictions, while the latter limits memory to sublinear in the input size but allows an edge's color to be announced any time after its insertion. We aim for the best of both worlds by designing small-space online algorithms for edge coloring. We study the problem under both (adversarial) edge arrivals and vertex arrivals. Our results significantly improve upon the memory used by prior online algorithms while achieving an $O(1)$-competitive ratio. In particular, for $n$-node graphs with maximum vertex-degree $Δ$ under edge arrivals, we obtain an online $O(Δ)$-coloring in $\tilde{O}(n\sqrtΔ)$ space. This is also the first W-streaming edge-coloring algorithm using $O(Δ)$ colors (in sublinear memory). All prior works either used linear memory or $ω(Δ)$ colors. We also achieve a smooth color-space tradeoff: for any $t=O(Δ)$, we get an $O(Δt (\log^2 Δ))$-coloring in $\tilde{O}(n\sqrt{Δ/t})$ space, improving upon the state of the art that used $\tilde{O}(nΔ/t)$ space for the same number of colors (the $\tilde{O}(.)$ notation hides polylog$(n)$ factors). The improvements stem from extensive use of random permutations that enable us to avoid previously used colors. Most of our algorithms can be derandomized and extended to multigraphs, where edge coloring is known to be considerably harder than for simple graphs.