Minimal dark energy models, described by the same number of free parameters of the standard cosmological model with cold dark matter plus a cosmological constant to parameterize the dark energy component, constitute very appealing scenarios which may solve long-standing, pending tensions. On the one hand, they alleviate significantly the tension between cosmological observations and the presence of one sterile neutrino motivated by the short-baseline anomalies: for some cases, the 95%CL cosmological bound on the mass of a fully thermalized fourth sterile neutrino is relaxed to ms < 1.3 eV. Interestingly, this limit is in agreement with the observations at short-baseline experiments. On the other hand, the Hubble tension is satisfactorily solved in almost all the minimal dark energy schemes explored here. These phenomenological scenarios may therefore shed light on differences arising from near and far universe probes, and also on discrepancies between cosmological and laboratory sterile neutrino searches.