Compress a sparse matrix
MATL, 19 bytes

!3#f!Dx0Gg!XsYshDq!


Input uses ; as row separator.

Try it online! Or verify all test cases: 1, 2, 3, 4, 5.

Explanation

!     % Implicit input. Transpose
3#f   % 3-output version of find: it takes all nonzero values and pushes
      % their column indices, row indices, and values, as column vectors
!     % Transpose into a row vector
D     % Display (and pop) vector of values
x     % Delete vector of row values
0     % Push 0
G     % Push input
g     % Convert to logical: nonzeros become 1
!     % Transpose
Xs    % Sum of columns. Gives a row vector
Ys    % Cumulative sum
h     % Prepend the 0 that's below on the stack
D     % Display (and pop) that vector
q     % Subtract 1 from the vector of row indices
!     % Transpose into a row vector. Implicitly display