Online Convex Optimization (OCO) is a powerful framework for sequential prediction, portraying the natural uncertainty inherent in data-streams as though the data were generated by an almost limitless adversary. However, this view, which is often too pessimistic for real-world data, comes with a price. The complexity of solving many important online tasks in this adversarial framework becomes much worse than that of their offline counterparts. In this work we consider a natural random-order version of the OCO model, in which the adversary can choose the set of loss functions, but does not get to choose the order in which they are supplied to the learner; Instead, they are observed in uniformly random order. While such a model is clearly not suitable for temporal data, which inherently depends on time, it is very much plausible in distributed settings, in which data is generated by multiple independent sources, or streamed without particular order. Focusing on two important families of online tasks, one which generalizes online linear and logistic regression, and the other being online PCA, we show that under standard well-conditioned-data assumptions (that are often being made in the corresponding offline settings), standard online gradient descent (OGD) methods become much more efficient in the random-order model. In particular, for the first group of tasks which includes linear regression, we show that OGD guarantees polylogarithmic regret (while the only method to achieve comparable regret in the fully-adversarial setting is the Online-Newton Step method which requires quadratic memory and at least quadratic runtime). This result holds even without assuming the convexity of individual loss functions. In the case of online k-PCA, we show that OGD minimizes regret using only a rank-k SVD on each iteration and requires only linear memory (instead of nearly quadratic memory and/or potentially high-rank SVDs required by algorithms for the fully-adversarial setting).