Dinucleoside tetraphosphates, often described as alarmones because their cellular concentration increases in response to stress, have recently been shown to function in bacteria as precursors to nucleoside tetraphosphate (Np4) RNA caps. Removal of this cap is critical for initiating 5' end-dependent degradation of those RNAs, potentially affecting bacterial adaptability to stress; however, the predominant Np4 decapping enzyme in proteobacteria, ApaH, is inactivated by the very conditions of disulfide stress that enable Np4-capped RNAs to accumulate to high levels. Here, we show that, in Escherichia coli cells experiencing such stress, the RNA pyrophosphohydrolase RppH assumes a leading role in decapping those transcripts, preferring them as substrates over their triphosphorylated and diphosphorylated counterparts. Unexpectedly, this enzyme recognizes Np4-capped 5' ends by a mechanism distinct from the one it uses to recognize other 5' termini, resulting in a one-nucleotide shift in substrate specificity. The unique manner in which capped substrates of this kind bind to the active site of RppH positions the δ-phosphate, rather than the β-phosphate, for hydrolytic attack, generating triphosphorylated RNA as the primary product of decapping. Consequently, a second RppH-catalyzed deprotection step is required to produce the monophosphorylated 5' terminus needed to stimulate rapid RNA decay. The unconventional manner in which RppH recognizes Np4-capped 5' ends and its differential impact on the rates at which such termini are deprotected as a prelude to RNA degradation could have major consequences for reprogramming gene expression during disulfide stress.