Depth First Search to find Minimum spanning tree
No you wont. Suppose you have as graph the complete graph $K_n$ with $n\ge 4$, for simplicity we pick the $K_4$, but this construction works for all larger $n$. 

Every DFS-tree of a complete graph is a path, no matter how you order the edges. But clearly you can define the weights such that a non-path is the MST. See the example below (MST in red).