In addition to Durack's heroics, David Parker's side also hit the bar twice in entertaining encounter at the Arriva Stadium in Crosby.
Straight from the kick-off, Birmingham threatened as Kirsty Linnett found space down the right and sent in an awkward low cross, which Durack did well to collect.
In the tenth minute a Chelsea Weston cross deceived Durack and struck the crossbar. As Rachel Williams tried to force the rebound home, Vicky Jones and Lindsay Johnson combined to prevent what seemed a certain goal.
As Birmingham continued to dominate, Jade Moore was unlucky when rattling the bar from 20 yards and, as the ball bounced down, Williams headed the rebound over.
Approaching the half-hour, it was Everton’s turn to hit the frame of the goal with Michelle Hinnigan’s first-time shot coming back off the bar.
Shortly before half-time Durack had to rush out of her goal as Jo Potter broke clear of the Everton defence. The England Under-19 international was shown a yellow card for blocking Potter on the edge of the penalty area.
From the resulting free-kick, Potter’s fierce shot was tipped over impressively by Durack.
The visitors began the second half as they finished the first, on the offensive, but again Durack denied them, acrobatically tipping over a shot by Linnett.
Everton began to establish a foothold in the game and in the 63rd minute Duggan set Brooke Chaplen free down the right, but Marie Hourihan narrowed the angle and the midfielder’s clipped effort passed by the face of the goal.
With 20 minutes to go Everton substitute Nikita Parris missed the best chance of the game, heading over an unguarded net from a corner.
Parris made a big impact after coming on and could have snatched a late winner when she met Jones’s excellent cross but her header was straight at Hourihan.
In the latter stages the home team continued to press, Simone Magill, another substitute, feeding Duggan whose toe-poke rolled wide. In the dying seconds of injury time Duggan had another excellent chance but again couldn’t find the target.
Birmingham coach Parker was delighted by his team’s performance, believing Durack prevented them notching a first Super League win of the season. “You know your own team have played well when the home keeper picks up the player of the match award," he said.
“Another night we could have gone in 3-0 at half-time and I don’t think anybody could have complained.”
Blues Ladies: Marie Hourihan, Chelsea Weston, Kerys Harrop, Laura Bassett, Rachel Unitt, Isobel Christiansen, Jo Potter, Jade Moore, Melissa Lawley, Rachel Williams, Kirsty Linnett (Kate Wilkinson 75).
Subs (not used): Emily Simpkins, Cristina Torkildsen, Hannah Gordon, Emily Westwood, Mary Earps, Marie Ballard
Bookings: Jade Moore 18, Rachel Williams 80.