No. You can measure curvature without reference to any higher-dimensional space that the curved surface might be embedded in - so the fact you can measure curvature by measuring angles in triangles doesn't necessarily mean that the surface is embedded in a higher dimensional space.

Spacetime is a four dimensional structure (a "manifold"). Space at one instant of time is a three dimensional "slice" through that four dimensional manifold. There is a lot of flexibility in how you choose to do the slicing, so "space" is a lot more subjective than spacetime. However, specifically in the kinds of spacetimes used in cosmology, there's only one sensible way to do the slicing and the geometry of the slices you get (flat, closed, or open) turns out to depend on the densities of matter and other stuff like radiation and dark energy.

Space, therefore, is a 3d manifold embedded in a 4d manifold. Spacetime is a 4d manifold that isn't embedded in anything as far as we know. Both can be curved, but it turns out that spacetime is curved and space (at least in cosmological models and cosmological conventions) isn't.

So to answer your question, depending on whether you meant space or spacetime when you wrote "universe", the universe is either three or four dimensional. If you meant space, yes it's part of a 4d structure, but if you meant spacetime then no it's not embedded in any higher dimensional space. Either way, the curvature or lack thereof doesn't imply anything about higher dimensions.

Sorry that's such a delightful mess of a yes-and-no.

(Note: there are a lot of theories that do require more than four dimensions, but we don't know which of them, if any, is correct. My answers are all based on standard GR cosmology - anybody telling you about 5+ dimensions is talking about much more speculative theories, however confident they sound.)