There are a few methods that are used to control erosion before permanent vegetation is established. The exact method used can depend on location (also country), season, preference, governing laws, etc.
This is almost always done. Mulch is frequently straw but can also be ground up newspaper. This helps to initially keep the soil from eroding while plants are growing. It also somewhat helps retain moisture in the soil so that plant will grow easier.
The mulch is sometimes pressed into the soil with tracked equipment, or a type of netting is placed with the mulch. Both of these help to keep the mulch from blowing away.
Hydro mulching is also a method that is frequently combined with hydroseeding.
The choice of seed is also a factor. Many times a portion of the seed used is a variety specifically chosen because it is quick growing. This plant may not be a species that will survive the winter, but it serves as a cover crop while the permanent vegetation grows.
In addition to the effects of mulch, other measures can be taken to slow down the speed of water flowing down a slope. Slow moving water is more likely to soak into the soil. Water that doesn't soak in will be moving slow enough that it won't cause erosion.
A few of these methods are:
Grade the top of slope so that rain water isn't funneled to one point. Counterintuitively, this means making sure that there isn't a small berm at the top of slope. A small berm will eventually fail and cause water to be directed to one spot, causing erosion. Without a berm, any water trying to flow down the slope is more evenly distributed.
Install terraces or fiber rolls staked across the slope. These help to keep the water from flowing in a straight line down the slope and causing erosion.
Mats or wattles of natural material can be placed. These function similarly to mulch, but may withstand higher water velocities in areas likely to flood.
Rock of various sizes can be placed. Plants will eventually grow through the rock.
It is usually not recommended, but in the past, an asphalt emulsion was sprayed on slopes to keep them from eroding. The thought was that plants would take root as the asphalt formed cracks.