Dryland vegetation is known for its ability to exhibit high degrees of spatial organization with well-defined patterns of vegetated and unvegetated soil patches. Deterministic dynamics are usually invoked to explain the emergence of these patterns in arid ecosystems, while the effect of noise on the process of pattern formation has seldom been investigated. Relatively strong interannual rainfall fluctuations are a typical feature of dryland climates. Rainfall variability is commonly interpreted as a source of random disturbances for arid and semiarid vegetation, while its ability to play a more fundamental role in the process of vegetation pattern formation remains poorly understood. Here we show how random interannual climate fluctuations may indeed cause the emergence of vegetation patterns through the random alternation of stressed (i.e., water limited) and unstressed conditions. Thus, patterns emerge as a noise-induced phenomenon from a mechanism of climate-driven random switching of deterministic dynamics. Moreover, pattern formation is observed even when the deterministic dynamics of unstressed and stressed vegetation would - separately - lead to uniform spatial distributions of vegetation or bare soil.