We present UniRank, a multi-agent LLM pipeline that estimates university positions across global ranking systems using only publicly available bibliometric data from OpenAlex and Semantic Scholar. The system employs a three-stage architecture: (a) zero-shot estimation from anonymized institutional metrics, (b) per-system tool-augmented calibration against real ranked universities, and (c) final synthesis. Critically, institutions are anonymized -- names, countries, DOIs, paper titles, and collaboration countries are all redacted -- and their actual ranks are hidden from the calibration tools during evaluation, preventing LLM memorization from confounding results. On the Times Higher Education (THE) World University Rankings ($n=352$), the system achieves MAE = 251.5 rank positions, Median AE = 131.5, PNMAE = 12.03%, Spearman $ρ= 0.769$, Kendall $τ= 0.591$, hit rate @50 = 20.7%, hit rate @100 = 39.8%, and a Memorization Index of exactly zero (no exact-match zero-width predictions among all 352 universities). The systematic positive-signed error (+190.1 positions, indicating the system consistently predicts worse ranks than actual) and monotonic performance degradation from elite tier (MAE = 60.5, hit@100 = 90.5%) to tail tier (MAE = 328.2, hit@100 = 20.8%) provide strong evidence that the pipeline performs genuine analytical reasoning rather than recalling memorized rankings. A live demo is available at https://unirank.scinito.ai .