Web servers provide content to users, with the requirement of providing high response quality within a short response time. Meeting these requirements is challenging, especially in the event of load spikes. Meanwhile, we observe that a response to a request can be adapted or partially executed depending on current resource availability at the server. For example, a web server can choose to send a low or medium resolution image instead of sending the original high resolution image under resource contention. In this paper, we exploit partial execution to expose a trade off between resource consumption and service quality. We show how to manage server resources to improve service quality and responsiveness. Specifically, we develop a framework, called Quota-based Control Optimization (QACO). The quota represents the total amount of resources available for all pending requests. QACO consists of two modules: (1) A control module adjusts the quota to meet the response time target. (2) An optimization module exploits partial execution and allocates the quota to pending requests in a manner that improves total response quality. We evaluate the framework using a system implementation in the Apache Web server, and using a simulation study of a Video-on-Demand server. The results show that under a response time target, QACO achieves a higher response quality than traditional techniques that admit or reject requests without exploiting partial execution.