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The exponential distribution is one of the most popular choices of model, especially in survival analysis and reliability theory. Typically, to test the hypothesis that the underlying distribution of lifetimes are exponentially distributed based on censored data, the standard goodness‐of‐fit tests used in the complete sample case are modified. However, none of the characteristic or Laplace function based tests have been adapted to incorporate censored data. In this paper, we modify three well‐known tests for exponentiality on the basis of either the characteristic function or Laplace transform to accommodate Type II right censored data. We compare the power of the three newly changed tests to some existing modified tests in an independent identically distributed (i.i.d.) and a model‐based setup through a Monte Carlo study, where it was found that the newly adjusted Baringhaus and Henze test outperformed all other tests considered