Yea, we used plastic webbing also and cable tied it to the fencing top rail and lapped it down so they couldn't go under and on the gate itself, but had to use a door sweep on the actual gate too. There was enough space for Sophie to get thru without even slowing down so we got creative with the aluminum door sweep vertically between the gate and the jam part.
You have get to get up early to out smart these Yorkies!