We address online bandit learning of Nash equilibria in multi-agent convex games. We propose an algorithm whereby each agent uses only obtained values of her cost function at each joint played action, lacking any information of the functional form of her cost or other agents' costs or strategies. In contrast to past work where convergent algorithms required strong monotonicity, we prove that the algorithm converges to a Nash equilibrium under mere monotonicity assumption. The proposed algorithm extends the applicability of bandit learning in several games including zero-sum convex games with possibly unbounded action spaces, mixed extension of finite-action zero-sum games, as well as convex games with linear coupling constraints.