We investigate odd-frequency superconducting correlations in normal-superconductor (NS) and short superconductor-normal-superconductor (SNS) junctions with the S region described by the Kitaev model of spinless fermions in one dimension. We demonstrate that, in both the trivial and topological phases, Andreev reflection is responsible for the coexistence of even- and odd-frequency pair amplitudes in N and S close to the interfaces, while normal reflections additionally only contributes to odd-frequency pairing in S. In the S region close to the NS interface we find that the odd-frequency pair amplitude exhibits large, but finite, values in the topological phase at low frequencies due to the emergence of a Majorana zero mode at the interface which also spreads into the N region. We also show that in S both the local density of states and local odd-frequency pairing can be characterized solely by Andreev reflections deep in the topological phase. Moreover, in the topological phase of short SNS junctions, we find that both even- and odd-frequency amplitudes capture the emergence of topological Andreev bound states. For a superconducting phase difference 0<φ<π the odd-frequency magnitude exhibits a linear frequency (∼|ω|) dependence at low frequencies, while at φ=π it develops a resonance peak (∼1/|ω|) due to the protected Majorana zero modes.