The home side were able to rest a number of players for the first round tie and prevailed at a canter thanks to lethal displays from Shannon McShane and Jess Walley.
A relentless second half, in which the hosts built on a 4-0 half-time lead with 11 further goals, did the damage at the club’s Community Pitch.
McShane stuck on four occasions with Walley also bagging a hat-trick as the favourites, who operate above Dawley in the women’s football pyramid, progressed to the next round.
Midfielder Walley got things going for her side, striking from inside the box after just six minutes.
Zoe Child soon got in on the act and made it two for Shrewsbury before McShane and Child each scored again before half-time.
Town were frustrated not to have more of a lead going in at the interval but quickly but that behind them in the second period.
And Dawley were undone twice just two minutes into the second half as Paris Schofield and Walley, with her second of the day, made it six.
McShane and Town were rampant from there on in. She clocked up a hat-trick with some excellent finishing while skipper Vikki Owen powered in a header from a corner.
Mia Pugh also impressed for Shrews, and got herself on the scoresheet as well as weighing in with three assists for her team-mates.
Substitute Libby Veitch played just 10 minutes at the end of the hammering but added to the scoreline with two goals of her own while Walley claimed a hat-trick.
Lottie Etterley was Shrewsbury’s other goalscorer on an afternoon to remember where spirited Dawley continued to battle until the end.
Boss Tom Peevor said: “I’m pleased, we played well, it was a game that provided a different challenge for us.
“We played quite a newly-formed team where we’ve been a team together for a lot longer.
“There was a clear difference between the teams, we had to show respect by not giving up and continuing as much as we could and fair play to Dawley for not giving up at all, the attitude from them was first class.”
It is not yet known when the draw will take place for the next round. Wem Town Ladies will also be in the hat after a heavy 6-0 win at AFC Telford a fortnight ago.
Shrewsbury, who dropped their first West Midlands League points of the season last time out against rivals Shifnal Town Ladies, resume their league campaign on Sunday as Peevor’s second-placed side go to Port Vale (2pm).
Leaders Shifnal, also unbeaten, are a point better off having played one game more at the summit.
They were not in action last weekend but face a tough trip to fifth-placed Sandwell Ladies on Sunday.
AFC Telford United Ladies have not been in league action for more than a month after falling victims to an away walkover last month following a mix-up in venue, for which they were also docked points.
Chris Brown’s side have slipped to sixth as a result but can climb above Sandwell by beating third-bottom Walsall Wood at the New Bucks Head on Sunday (2pm).
In the National League Division One Midlands, Wem Town were scheduled to travel to Burton Albion but the fixture was rained off.
Wem are not in action this weekend and return on Sunday week at Peterborough United.
The New Saints Ladies picked up a huge three points courtesy of 2-0 home victory over Port Talbot Town, which lifted them off bottom spot in the Adran Premier.
Caitlin Chapman and Lexie Harrison struck inside the final 10 minutes of the first half to hand the hosts just a second home league win of the season. TNS managed the second half well to see out the crucial Park Hall success.
The Saints face a difficult task at leaders Swansea City, who are unbeaten with 11 wins from 12 this season, on Sunday.