A major goal of computational neuroscience is to develop powerful analysis tools that operate on large datasets. These methods provide an essential toolset to unlock scientific insights from new experiments. Unfortunately, a major obstacle currently impedes progress: while existing analysis methods are frequently shared as open source software, the infrastructure needed to deploy these methods -at scale, reproducibly, cheaply, and quickly -remains totally inaccessible to all but a minority of expert users. As a result, many users can not fully exploit these tools, due to constrained computational resources (limited or costly compute hardware) and/or mismatches in expertise (experimentalists vs. large-scale computing experts). In this work we develop Neuroscience Cloud Analysis As a Service (NeuroCAAS): a fully-managed infrastructure platform, based on modern large-scale computing advances, that makes state-of-the-art data analysis tools accessible to the neuroscience community. We offer NeuroCAAS as an open source service with a drag-and-drop interface, entirely removing the burden of infrastructure expertise, purchasing, maintenance, and deployment. NeuroCAAS is enabled by three key contributions. First, NeuroCAAS cleanly separates tool implementation from usage, allowing cutting-edge methods to be served directly to the end user with no need to read or install any analysis software. Second, NeuroCAAS automatically scales as needed, providing reliable, highly elastic computational resources that are more efficient than personal or lab-supported hardware, without management overhead. Finally, we show that many popular data analysis tools offered through NeuroCAAS outperform typical analysis solutions (in terms of speed and cost) while improving ease of use and maintenance, dispelling the myth that cloud compute is prohibitively expensive and technically inaccessible. By removing barriers to fast, efficient cloud computation, NeuroCAAS can dramatically accelerate both the dissemination and the effective use of cutting-edge analysis tools for neuroscientific discovery.