After eight minutes, Brentford led. Christian Norgaard, making his 100th Premier League appearance, spread a pass to the right wing for Mbeumo who cut inside. His shot flicked off Fabian Schar and past the static Pope.
But four minutes later, Newcastle equalised when Jacob Murphy's speculative shot from 25 yards was pounced on by Isak flinging himself in the way to head home.
Newcastle had the better chances to take the lead but were punished for not taking them when Barnes gave the ball away in his own half with a loose pass. Wissa picked it up, ran at the retreating defence and curled a fine finish past Pope from the edge of the area.
Barnes made amends in the 32nd minute, allowed too much time and space in the Brentford area to turn and finish low through the legs of Norgaard and past Mark Flekken at the far post.
But the winner came in the 56th minute when Newcastle's defence allowed a long set-piece from Flekken to bounce in the area, with Collins stroking his second goal of the season across Pope.
And the victory was sewn up late on when Mbeumo fed Schade who cleverly clipped his shot over the advancing Pope for 4-2.