If I can solve Sudoku, can I solve the Travelling Salesman Problem (TSP)? If so, how? Let us say there is a program such that if you give a partially filled Sudoku of any size it gives you corresponding completed Sudoku.

Can you treat this program as a black box and use this to solve TSP? I mean is there a way to represent TSP problem as partially filled Sudoku, so that if I give you the answer of that Sudoku, you can tell the solution for TSP in polynomial time? 

If yes, how? how do you represent TSP as a partially filled Sudoku and interpret corresponding filled Sudoku for the result.
For 9x9 Sudoku, no.  It is finite so can be solved in $O(1)$ time.

But if you had a solver for $n^2 \times n^2$ Sudoku, that worked for all $n$ and all possible partial boards, and ran in polynomial time, then yes, that could be used to solve TSP in polynomial time, as completing a $n^2 \times n^2$ Sudoku is NP-complete.  

The proof of NP-completeness works by reducing from some NP-complete problem R to Sudoku; then because R is NP-complete, you can reduce from TSP to R (that follows from the definition of NP-completeness); and chaining those reductions gives you a way to use the Sudoku solver to solve TSP.