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I remember Maltsiniotis is thanked in the "Préambule" of Cisinski's book: « en particulier, il a amélioré la définition d'extension anodine en dégageant une axiomatique simple ».
Dear Harry: The link to Cisinski's publications page is math.univ-toulouse.fr/~dcisinsk/publications.html. The fifth link points towards a copy of the book: math.univ-toulouse.fr/~dcisinsk/ast.pdf. Anodyne extensions in this general setting were already mentioned in "Théories homotopiques dans les topos" (second link of the publications page) but since I'm not familiar with the theory I don't know whether it was their first occurrence in the literature.
