While recent advances in large language models (LLMs) have significantly enhanced performance across diverse natural language tasks, the high computational and financial costs associated with their deployment remain substantial barriers. Existing routing strategies partially alleviate this challenge by assigning queries to cheaper or specialized models, but they frequently rely on extensive labeled data or fragile task-specific heuristics. Conversely, fusion techniques aggregate multiple LLM outputs to boost accuracy and robustness, yet they often exacerbate cost and may reinforce shared biases. We introduce EMAFusion, a new framework that self-optimizes for seamless LLM selection and reliable execution for a given query. Specifically, EMAFusion integrates a taxonomy-based router for familiar query types, a learned router for ambiguous inputs, and a cascading approach that progressively escalates from cheaper to more expensive models based on multi-judge confidence evaluations. Through extensive evaluations, we find EMAFusion outperforms the best individual models by over 2.6 percentage points (94.3% vs. 91.7%), while being 4X cheaper than the average cost. EMAFusion further achieves a remarkable 17.1 percentage point improvement over models like GPT-4 at less than 1/20th the cost. Our combined routing approach delivers 94.3% accuracy compared to taxonomy-based (88.1%) and learned model predictor-based (91.7%) methods alone, demonstrating the effectiveness of our unified strategy. Finally, EMAFusion supports flexible cost-accuracy trade-offs, allowing users to balance their budgetary constraints and performance needs.