Generating finite simple groups with $2$ elements

The answer to your question is yes. Moreover, if you pick two random elements from a finite simple group, then they generate the whole group with probability which tends to 1 as the size of the group grows. There are even stronger results in this direction, but I am not an expert in the subject so you will have to look for it yourself. You should look for papers by Liebek and Shalev, Lubotzky, Kantor, and there are others who I am not sure about now.

All of these results require the classification. There are very few results regarding finite simple groups which do not require the classification.

Edit: here is a link to a fairly old survey paper in the notices: http://www.ams.org/notices/200104/fea-shalev.pdf. There are many new developments in the last decade.

Since I happen to know the OP is number-theoretically inclined, let me add the following remark:

For "most" finite simple groups $G$ it is indeed the case that $G = \langle x, y \rangle$ where $x$ has order $2$ and $y$ has order $3$. Equivalently, $G$ is a quotient of the free product $\mathbb{Z}/2\mathbb{Z} * \mathbb{Z}/3\mathbb{Z} \cong \operatorname{PSL}_2(\mathbb{Z}) = \Gamma(1)$.

This has the following geometric consequence: there is some subgroup $\Gamma_G \subset \Gamma(1)$ such that $X_G = \Gamma_G \backslash \overline{\mathcal{H}}$ is a modular curve and $X_G \rightarrow X(1) \cong \mathbb{P}^1$ is a $G$-Galois branched covering. By taking $G$ to be something else than $\operatorname{PSL}_2(\mathbb{Z}/p\mathbb{Z})$ one sees that $\Gamma(1)$ admits many non-congruence subgroups. For instance, it is well-known (added: I should have said "a well-known theorem of J.G. Thompson") that one can take $G$ to be the Fischer-Griess Monster.

I don't want to make precise what I mean by "most". Note that there are infinitely many finite simple groups with order prime to $3$ (although one has to look fairly far down the list of all finite simple groups to see them: Suzuki groups), so I definitely do not mean "all but finitely many".

In addition to the two answers already given it might be worth to mention that the generating graph of a finite simple group has no isolated vertices: This means that for every nonidentity element $x\in G$, there is some other element $y$ such that $G=\langle x, y\rangle$. (The generating graph of a group has the nonidentity elements of $G$ as vertices, where to vertices are connected if they generate the group.) This is shown in

Guralnick, Robert, Kantor, William, Probalistic generation of finite simple groups, J. Algebra 234 (2000), p. 743–792. (MR1800754)

Recently, Breuer, Guralnick, Lucchini, Maróti and Nagy have shown that the generating graph of every "sufficiently large" finite simple group contains a Hamiltonian cycle. You might also look at the references given in their paper:

Breuer, T., Guralnick, R. M., Lucchini, A., Maróti, A., Nagy, G. P., Hamiltonian cycles in the generating graphs of finite groups, Bull. Lond. Math. Soc. 42 (2010), p. 621–633. (MR2669683)