Can we have a one-one function from [0,1] to the set of irrational numbers? Since both of them are uncountable sets, we should be able to construct such a map. Am I correct?
If so, then what is the map?

There are lots of good answers already, but I thought I might add one more using a flexible method that shows up a lot in descriptive set theory.
Note that $\mathbb{R} \setminus \mathbb{Q}$ is the intersection of countably many dense open sets $(G_n : n \in \mathbb{N})$; just take $G_n = \mathbb{R} \setminus \{q_i\}$ for some enumeration $(q_i : i \in \mathbb{N})$ of the rationals.
Note also that there is an injection from $[0,1]$ to the Cantor space $2^\mathbb{N}$ given by binary expansions.  If there is a choice between a binary expansion that is eventually zeroes and another that is eventually ones, take the one that is eventually zeroes. (This injection is not continuous.) 
So it suffices to prove the following general fact:

Let $(G_n : n \in \mathbb{N})$ be a sequence of dense open subsets of $\mathbb{R}$.  Then there is an injection from $2^\mathbb{N}$ to $\bigcap_{n \in \mathbb{N}} G_n$. (As we will see, there is a continuous injection.)

By recursion, we can define for each finite sequence $s$ of natural numbers a closed interval $I_s$ satisfying the following properties:


*

*$I_s$ has length between zero and $2^{-n}$ where $n$ is the length of the finite sequence $s$,

*$I_t \subset I_s$ whenever $t$ extends $s$,

*$I_s \cap I_t = \emptyset$ whenever $s$ and $t$ are incompatible, and

*$I_s \subset G_n$ where $n$ is the length of $s$.
This binary tree of intervals is called a Cantor scheme.  If desired, we can define $I_s$ explicitly as $[q_i,q_j]$ where $(i,j)$ is the lexicographically least pair of natural numbers making this interval obey the rules.
Every element $(d_i : i \in \mathbb{N})$ of the Cantor space $2^\mathbb{N}$ determines a branch in this binary tree.  Our continuous injection is obtained by mapping $(d_i : i \in \mathbb{N})$ to the unique point in the intersection of the corresponding sequence of closed intervals 
$$I_{d_0} \supset I_{d_0,d_1} \supset I_{d_0,d_1,d_2} \supset \cdots.$$
