Consider (x,y) in R2 with x,y both in (0,1). Write x as some decimal $x=0 .a_1a_2a_3...$ and $y=0.b_1b_2b_3b_4...$
Now, write z in R as $a_1b_1a_2b_2a_3b_3...$.
If y and x are both finite, pad the one of lower length with zeros until they are of the same length. If x is finite and y is infinite, pad x with infinite zeros.
This is a bijection because each x, y maps uniquely onto some x in R, and from each x one can derive x,y.
The straightforward digit-interleaving you describe is not surjective. The numbers $$ \frac{227}{660} = 0.34393939\ldots \qquad\text{and}\qquad \frac{233}{660} = 0.35303030\ldots $$ cannot both be in the range, because the only pair that can map to either is $(\frac13,\frac12)$.
A trick sometimes used to get around this is to interleave not single digits, but "blocks", where a block consists of any finite number (possibly zero) of $9$s, followed by a digit that differs from $9$. There is exactly one way to represent each number in $[0,1)$ as an infinite sequence of blocks, so interleaving the blocks does produce an actual bijection $[0,1)^2\to[0,1)$.