In my experience it is the opposite...
Gradient descent is the basic minimization algorithm and for large problems is often unusable because the full gradient calculation is too "expensive" to do every step or perhaps at all.
Many variations, including stochastic gradient descent, markov monte-carlo (simulated annealing), pocs, compressed sensing, etc. use some knowledge of the system or constraint to simplify the gradient calculation or get some information about what is the best direction to search for the answer (solution minimizing the error, energy, cost function or whatever one is minimizing).
Stochastic gradient descent calculates only one component (say out of M components) of the full gradient each iteration. Therefor each iteration of stochastic gradient descent costs (in operations) 1/M that of gradient descent.
It happens that for most problems the single component (or subset) gradient still reduces the error each iteration, so not that many more iterations are needed.
For a system with M degrees of freedom, stochastic gradient descent can be up to M times faster than gradient descent.
There is certainly much more to the discussion, including guarantees and rates of convergence, but the above is the gist.
For reference here is the Stanford cs221 lecture that discusses gradient descent. It continues into stochastic gradient descent.