LLM-as-a-Judge systems are increasingly used to evaluate text at scale, yet production deployment demands low latency, minimal cost, and compatibility with closed-source APIs. Current approaches fall short in different ways: some require many LLM calls and per-dataset prompt tuning, others depend on logit access unavailable in commercial APIs, and yet others demand multiple rounds of LLM interaction for iterative feature discovery. We present SAJA (Simple Approach to Judge Alignment), built on the principle that task-specific alignment should reside in a lightweight calibration head, not in elaborate prompts or model internals. SAJA makes exactly one LLM call per item using a fixed structured rubric prompt, extracts a multi-dimensional feature vector, and maps it to a human-aligned score via a calibration head trained on a small number of human labels. No iterative prompt search, no logit access, and no multi-round LLM interaction are needed. Yet SAJA matches far more complex systems across four evaluation paradigms: 86% F1 on MT-Bench pairwise preference (vs. 78% uncalibrated), competitive performance on five classification benchmarks with a single call, and +5.71% F1 over prompt-optimized baselines on proprietary data. Ablations confirm that multi-dimensional rubric features outperform one-dimensional calibration (SummEval 𝜌 improves from 0.60 to 0.74) and that coarse rubric outputs recover the same human alignment as full logit distributions (𝜌 = 0.36 vs. 0.37), establishing that logit access is unnecessary for calibrated judge alignment. Moreover, SAJA is model-agnostic: a 9B open-source model with SAJA (𝜌=0.70) surpasses raw GPT-4.1 (𝜌=0.60). Its single-call design yields up to 4.8× cost savings over per-question approaches.