It is well-known that the first time a stable subordinator reaches [1, +∞). is Mittag-Leffler distributed. These distributions also appear as limiting distributions in triangular Polya urns. We give a direct link between these two results, using a previous construction of the range of stable subordinators. Beyond the stable case, we show that for a subclass of complete subordinators in the domain of attraction of stable subordinators, the law of the first passage time is given by the limit of an urn with the same replacement rule but with a random initial composition.