I had the same problem on a small lawn in zone 5 about 12 years ago. I did
it in steps to make it less work but I suppose there must be a way to do it
larger scale (i.e. complete lawn).
The process begins in late summer for a fall planting of the new lawn.
I shallowly tilled the area in order to loosen the bulk of the more shallow
roots. Rake this layer off and compost it. Make sure the compost gets
REALLY hot if you are going to do this because you don't want to transport
the weed problem. If you compost at a high enough temperature most of the
weeds will not survive. I had the space also to solarize the top layer
after composting several times befor re-using it in the garden.
Now that the top layer is gone, plug the remaining bare soil and water quite
well this will introduce moisture more deeply in the soil and this moisture
is important for the next step.
Lay clear plastic (not black) on top of the remaining bare soil, anchor the
sides in a soil filled trench to hold in all of the heat from the sun that
you can. Obviously this will not work under trees and such but you asked
for a non-chemical way. Leave the plastic in place during the hottest part
of the season as you want the sun to solarize the soil as deep as you can.
I have heard of others who will remove the plastic, lightly till and replace
the plastic to get the heat to go deeper but I did not do it that way. I
left the plastic in place for three weeks then used a clean organic mulch
NOT grass clippings from a weedy lawn to tide me over until the temperatures
cooled enough at night to plant the new grass. I then top dressed the mulch
with 1/2 inch of soil and planted the seed.
Rake the seed lightly to cover it with soil then roll over it with an EMPTY
roller to assure good contact with the soil. I don't like the looks of
wheat straw so I had some pine straw brought up by a friend in the south.
Maybe you could find pine straw (actually baled pine needles) up there where
you live but I could not at that time. It is perfectly fine to use wheat
straw but I found that the pine straw stayed in place better during high
Care for the new grass as you would any other new lawn. It worked quite
well for me. I did a quarter of my yaard a year for four years the results
were very good that way but I overlapped so that I left no ground