Complex systems can undergo abrupt state transitions near critical points. Theory and controlled experimental studies suggest that the approach to critical points can be anticipated by critical slowing down (CSD), i.e., a characteristic slowdown in the dynamics. The validity of this indicator in field ecosystems, where stochasticity is important in driving transitions, remains unclear. We analyzed long-term data from a dryland ecosystem in the Shapotou region in China and show that it underwent an abrupt transition from a nearly bare to a moderate grass-cover state. Prior to the transition, the system showed no (or weak) signatures of CSD, but exhibited expected increasing trends in the variability of the grass cover, quantified by variance and skewness. These surprising results are consistent with the theoretical expectation of stochastically driven abrupt transitions that occur away from critical points; indeed, a driver of vegetation - annual rainfall - showed rising variance prior to the transition. Our study suggests that rising variability can potentially serve as a leading indicator of stochastically driven transitions in real world ecosystems.