The point is that all these concepts (triangulated cat, dg-cat, quasitriangulated dg-cat, A-oo cat) are really approximations and/or presentations of _stable (oo,1)-categories_ . Every triangulated category that appears in nature is the homotopy category of a stable (oo,1)-category. Hence working just with triangulated categories is wrong in just the same way as just working with derived categories is wrong: because all higher homotopies have been divided out all higer universal constructions such as homotopy limits and colimits have been lost.

See <a href="http://ncatlab.org/nlab/show/stable+(infinity%2C1)-category">nLab:stable (oo,1)-category</a> and references given there. See in particular the section on "alternative models".