According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his Methodus inveniendi , Chapter 2, no. 21. Quite often, one speaks of Lagrange's differential equation, or the Euler-Lagrange equations. Yet Lagrange himself attributes this equation to Euler: 'Cette équation est quelle qu'Euler a trouvée le premier' (Oeuvres , Vol. 10, p. 397)."
NB: Euler's Methodus inveniendi was published in 1744.
Further down the same page in G&H, one finds "With Lagrange's letter of August 12, 1755 to Euler, the development of the variational calculus took a new turn. Lagrange explained the $\delta$-symbolism which quickly leads to Euler's equation. Euler was very impressed by the new method, and already one year later he lectured about it at the Berlin Academy."