Quillen's construction of the cotangent complex used homotopical algebra to find the correct higher-categorical object without explicitly building a higher category. This may sound newfangled and modern, but if you read Grothendieck's book on the cotangent complex, his explicit higher-categorical construction was only able to build a cotangent complex that had its (co)homology truncated to degree 2. Strangely enough, by the time Grothendieck's book was published, it was already obsolete, as he notes in the preface (he says something about how new work of Quillen (and independently André) had made his construction (which is substantially more complicated) essentially obsolete).