Getting N top scores from a matrix
What you want to do is view your problem as a problem on a directed layered graph. This will allow you to solve the problem using algorithms for the $k$-th shortest paths problem. 

In your example you would construct this graph:


The arcs with no label have weight zero. 

Finding the $k$-th shortest paths from $s$ to $t$ will find the $k$ top scores according to your definition. There are many efficient algorithms for obtaining the $k$-th shortest paths, including a variation on Dijkstra's algorithm which is probably the easiest to implement. 

PS. Some of these algorithms require that the weights of the arcs be non-negative. That can be easily achieved in your case, by adding a large enough constant to every weight, e.g., shifting every weight by +13 in your example. The result is equivalent, since all the paths from $s$ to $t$ contain the same number of arcs.