Clustering is an unsupervised techniques (requiring no ground truth) while classification is a supervised technique (which requires labels).

Although, it is indeed not uncommon to perform some feature engineering using clustering techniques. As an example, you could create an extra feature that indicates in which cluster a sample belongs.

An example of practical application of clustering + classification is given in the below paper:

Ferraretti, D., Gamberoni, G., & Lamma, E. (2012). Unsupervised and supervised

learning in cascade for petroleum geology. Expert Systems with Applications,

39(2012), 9504–9514.

http://www.sciencedirect.com/science/article/pii/S0957417412003673?via%3Dihub

Basically, the authors followed the same idea reported in the answers of Dr. Ofoghi and Dr. Agnihotri 

The approach is suitable in situations when you have no class labels for a supervised classification. Clustering might provide them. However, sometimes you don't know how many classes exist. Thus, finding a correct number of clusters and a meaningful interpretation of the clusters is very important and often must be done manually.