We consider some deterministic nonlinear PDEs on the torus whose solutions may explode in finite time for initial data above some threshold. Under suitable conditions on the nonlinear term, we show that explosion is suppressed by multiplicative noise of transport type in a certain scaling limit. The main result is applied to the 3D Fisher-KPP and 2D Kuramoto-Sivashinsky equations, yielding long-time existence for large initial data with high probability.