The Scottish international was introduced as a second half substitute by Gareth Taylor and doubled our advantage on the night with a scintillating long-range effort.
Filippa Angeldahl had put the hosts ahead inside 20 minutes with an incisive low finish, as City dominated proceedings from the off.
And Janine Beckie rounded off the scoring with a clinical half-volley as stoppage time approached to secure a second win in two Group B matches.
City dominated possession in the early stages, confidently zipping the ball around our opponents, but were unable to breach the stubborn Durham rear guard.
But that changed on 15 minutes, when Beckie found some space down the right flank and cut the ball back into the path of the onrushing Angeldahl.
The Swedish international took a touch on the edge of the visiting box before confidently stroking home into the bottom corner for her second goal in City blue.
It was the spark that Gareth Taylor’s side needed to transfer our possessional dominance into genuine chances.
Beckie almost doubled our advantage ten minutes later when she deflected Alanna Kennedy’s header from a corner towards goal, but Durham right-back Ellie Christon cut the celebrations short with a goal line clearance.
City went close once again soon after, with Hayley Raso dispossessing Grace Ayre in the box thanks to some tenacious wing play, but her square ball to Ellen White was tipped away from danger by Megan Borthwick before our skipper could tap home.
And Beckie was a coat of paint from grabbing our second in spectacular fashion on the stroke of half-time, with her curling left-footed effort from the edge of the box crashing back off the Durham bar with Borthwick stranded.
Our overall dominance continued after the restart, with Beckie twice going close in the opening 15 minutes after being moved into a more advanced position.
The Canadian’s first sight of goal came when she was found by an incisive Keira Walsh pass on the edge of the Durham box but, after working the ball onto her left, her path to goal was thwarted by a vital last-ditch block from Ayre.
Beckie went close again on the hour when she got on the end of White’s deft ball into the six yard box but under close attentions from Christone, she was unable to sort her feet out in time to trouble Borthwick.
City were denied by the woodwork for a second time soon after, with Angeldahl a matter of inches from completing her brace after a clever one-two with Laura Coombs had opened a gap in the Durham defence.
In need of a further spark to find that elusive second, Gareth Taylor called Weir into the action with 20 to play, as the Scot racked up a century of appearances for the Club.
The midfielder marked the occasion with a sensational strike, firing into the roof of the net from the edge of the Durham box to double our advantage late on.
And Beckie got the goal her performance deserved as stoppage time approached, smashing home at the far post to put the result beyond doubt.
The victory ensures City remain top of Group B in the Conti Cup, and go into Sunday’s Women’s Super League clash away to Leicester in confident mood.
How we lined up
CITY XI: Taieb, Beckie, Mace, Kennedy, Stokes, Losada (Walsh 46’), Angeldahl, Coombs, Raso (Greenwood 46’), Park, White (C) (Weir 71’).
SUBS: Pilling, Scott, Hemp
DURHAM XI: Borthwich, Christon, Wilson (C), Holmes (Robson 46’), Ayre, Bradley, Greenwood (Lambert 66’), Brown (Hepple 58’), Johnson (Briggs 46’), Roberts (Nicholson 58’), Crosthwaite
SUBS: Mackain, Hill, Galloway, Salicki
Attentions turn back towards in Women’s Super League action at the weekend, with an away clash against Leicester City next on the cards for Gareth Taylor’s side.
The match takes place on Sunday 7 November, with kick-off scheduled for 14:00 (UK).
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