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Write $q=1-p$. A disconnected graph has a connected component of a graph of size $1\le s\le \lfloor n/2\rfloor$. For a fixed $s$-element vertex set $S$, all $s(n-s)$ edges between $S$ and its complement must be absent. Their independent absence has probability $q^{s(n-s)}$. The union bound givesSet $r_n=n{q}^{n/2}$. Since $p$ is fixed in $(0,1)$, $r_n\to 0$. For large $n$, the final sum is at most the geometric series $r_n/(1-r_n)$, which tends to zero. Hence the binomial random graph is connected with probability tending to one, as in connectivity of a fixed-density binomial random graph.