Holders Chelsea reached the Women's FA Cup semi-finals for a sixth consecutive season as Ji So-yun's goal saw them narrowly win at part-time side Durham.
South Korea forward Ji turned the ball home at the far post after a deep, first-half cross from Erin Cuthbert.
The Championship side had earlier seen Lisa Robertson's free-kick tipped on to the bar by England's Carly Telford.
Chelsea went close to a second after the break but Ramona Bachmann and Scotland's Cuthbert both hit the post.
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Durham, who won away at top-flight opposition in Bristol City in the previous round, caused the European quarter-finalists problems and almost levelled dramatically in stoppage time, but Zoe Ness' effort was cleared off the line.
The semi-professional outfit, who are fourth in the second tier with a squad including painters and bar staff, made the better start, but Robertson's speculative free-kick hit the woodwork.
They were backed by a new club-record crowd of 1,629 at a gusty New Ferens Park, against Emma Hayes' side who are six points off the top of the Women's Super League.
Chelsea are bidding to win the cup for a third time and defend the trophy they won with last year's 3-1 win over Arsenal at Wembley.
The draw for the semi-finals will be held on Monday, live on BBC Breakfast on BBC One, shortly after 08:30 GMT.
Durham captain Sarah Wilson: "We average 400 people, who come week-in, week-out, and it was brilliant to have such a big support. Every single person who supported us was fantastic.
"We had some great chances. We defended well from the front. We were so well-disciplined. To play as we did and hold a quality side like Chelsea to just 1-0 is something we can be really proud of.
"Chelsea had clearly done their homework on us. For a top WSL team like Chelsea to come here and change their players and tactics around is a great credit to us."
Chelsea midfielder Jess Carter: "It was very tough. The conditions probably worked in Durham's favour and they were a strong side.
"It got a bit shaky at the end but we got the 1-0 and that was all that mattered.
"With the wind as bad as it was we were unable to play the type of football that this Chelsea team wants to play, but it's football - you have to adapt. We got the goal and got the win. That's what counts."
Durham: Reid, Holmes (Gears 53), Robson, Robertson, Christon (Dixon 87), Wilson, Ness, Salicki, Roberts (Cottam 87), Hill, Hepple.
Subs not used: Borthwick, Knott.
Chelsea: Telford, Blundell (Bachmann 58), Bright, Carter, Ji, Riley, Kirby, England, Eriksson, Mjelde, Cuthbert (Engman 86).
Subs not used: Berger, Cooper, Andersson, Carney, Spence.
Referee: David Robinson