Let A and B be almost commuting (i.e, AB-BA∈S₁) self-adjoint operators. We construct a functional calculus ϕ↦ϕ(A, B) for ϕ in the Besov class B∞,11(R2). This functional calculus is linear, the operators ϕ(A, B) and ψ(A, B) almost commute for ϕ,ψ∈B∞,11(R2), ϕ(A,B)=u(A)v(B) whenever ϕ(s,t)=u(s)v(t), and the Helton-Howe trace formula holds. The main tool is triple operator integrals.