Is it correct to say H2SO4 behaves like an acid in this reaction?
In general, this depends on the definition of acid that you are using.

For this specific case, the Broensted definition would suffice.
A Broensted acid is a reagent that donates a proton ($\ce{H^+}$) to its reaction partner. 

The protons of the sulfuric acid are transfered (donated) to the carbonate anion, which gives carbonic acid. Carbonic acid decomposes into water and carbon dioxide, as you will experience when drinking sparkling water.

The key thing now is the affinity to donate said proton, which leads to the concept of acid strength and can be expressed in terms of $K_\mathrm{a}$ values. 

If the anions were reversed and $\ce{ZnSO_4}$ would react with $\ce{H_2CO_3}$, one would find no reaction occuring.

This is a very easy explanation of the things going on, and I'd suggest you read a bit into 


Acid definitions: Broensted and Lewis acids
Acid strength: Equilibrium of dissociation reactions
Conjugated Acid / Base pairs