Has this spot been mulched with woody material before? If not, I'd be more hesitant as well. With sawdust or any freshly shredded wood, the nitrogen robbing described above can happen in a spot that hasn't yet established a population of microbes that decompose woody material. But one has to start with something, sometime...
DH brings sawdust home from work occasionally, but I've never used it as a first or only mulch material. It's spread/slung around beds where decomposition of a myriad of organic matter is already happening on the surface.
If you're able to add grass from mower bag, equal amount of grass or more, I would be confident about it going well. You would want to mow before grass (or other lawn plants) has seed heads. Add more to the surface if it disappears while the sawdust still looks fresh (light in color.) Another option for mixing with the sawdust would be moist produce scraps. Pureรฉ first if you don't want to recognize the bits (which would also greatly speed the decomposition process, and aid in mixing with sawdust.)
If you choose to use the plain sawdust with nothing added, adding a few successive additions of thinner layers might be your best bet, vs. adding everything you have in one thicker layer. More than an inch or so of any single kind of OM at a single time can form into an at least temporarily unproductive mat, that can sometimes shed water instead of absorb it, until the decomposition process gets going well.
I try to maintain a cover of constantly decomposing stuff, from compost to leaves, to grass clippings, whatever presents itself that's OK to look at in a given spot (or sometimes slightly buried if not.) Mulch doesn't have to come from a bag at all.
