I actually completed this my first go with randoms and no damage to the door with Soldier 76, Torbjorn, Genji, and Zenyatta.
The challenge of this mission is if a bot gets close to you they will explode and damage you. This will be fatal at lower health.
How we did this was Torbjorn, Soldier 76 (me) and Zenyatta stayed near the door killing zomnics from a distance. Torbjorn should keep his turret on the ledge at the right of the door. Zenyatta would keep his healing orb on genji most of the the time while genji dashed around in the back and upper left ledge blowing up zomnics. If you do this as genji, just be careful not to get caught in several explosions at once as they will kill you fast. Soldier 76 could provide his biotic field healing for the three of you by the door if needed so zenyatta can focus on keeping genji alive. Additionally as Soldier it's good to use your alt either when you're getting overwhelmed by zomnics since you really don't want them getting close to you, or when the boss's spawn to clear the zomnics quick while the rest of the team focus on the boss. This way you won't have to worry about explosions around the already deadly boss's. Torbjorn should make sure he has his alt ready for a boss as that will do a lot of damage to them not having a McCree and Soldier is already on clear duty.
Obviously I'm sure there are many good synergies that can beat this mode, but this is the one we did and like I said it worked out good enough to not even take door damage.