Van Gaal tasted victory last weekend with a 2-1 win over West Ham, and today's game was equally nervy as Roberto Martinez's Toffees ran United all the way to the final whistle.
United took the lead in the first half when Juan Mata laid the ball back to Angel Di Maria, who swept the ball past Tim Howard from the edge of the box to score his third goal since joining from Real Madrid.
But Martinez's Everton wouldn't lie down, and missed a penalty when Luke Shaw tripped Tony Hibbert inside the box, with Leighton Baines failing to beat David De Gea in the United goal from 12 yards.
Baines did have a hand in the equaliser though, crossing the ball to the back post, where Steven Naismith nodded past de Gea to draw Everton level.
However, United claimed all three points when Radamel Falcao grabbed his first goal in a Manchester United shirt, turning home Di Maria's goalbound shot from just inside the box.
United needed three world class saves from De Gea in the final minutes to stay in front, with the Spanish international keeping a number of Everton players at bay to hand United all three points.