The 30-year-old had unwittingly opened the scoring for Pep Guardiola's men right on the stroke of half-time, acrobatically vollying Jesus Navas's cross into his own net.
Eddie Howe's team doubled their advantage when Jose Sa and Max Kilman failed to deal with Jacob Murphy's low cross and Gordon pounced on the loose ball to fire home.
Ten Hag's gamble in dropping both Marcus Rashford and Anthony Martial amid reports of dressing-room disharmony had the desired effect for the under-fire manager, with the hosts' quality with and without the ball a world away from Saturday's
Munir El Haddadi, starting for Suarez, doubled the lead from Messi's cross and the Argentine played in Neymar to score the third as Barca breezed through the visiting defense at will.