The Blues striker struck her first goal of the season in injury time after coming off the bench, but her effort proved only to be consolation as the Citizens claimed victory in the opening Women’s League Cup game.
Former Everton loanee Jess Park gave the visitors the lead after 20 minutes before Khadija Shaw made it two just after the restart.
Duggan’s header gave the crowd some cheer, the 32-year-old looked razor sharp when meeting Heather Payne’s cross to give her side a deserved goal.
And the strike - her first since returning to action from maternity leave - is something she is hoping to build on as Everton head into a fixture the forward believes is the perfect opportunity to turn things around.
“It’s my job to score goals, so it’s nice to get off the mark,” she said. “It hinders it a little bit with the result, it was unfortunate we didn’t get the result we wanted.
“On a personal level, I’m really happy to help the team more than anything. Hopefully more to come.”
Turning her attention to Sunday, Duggan continued:
“There’s no better way to turn things round than winning the Merseyside derby, “Liverpool are our rivals. They always have been.
“It’s a massive game, but you don’t want to ride the occasion. We just have to see as three points; we’ve got to get off the mark in the league.
“We have a great team, great quality players. It’s been a slow start, but we know what we’re capable of, we just need to show it on the pitch.
“Anfield, it’s on my doorstep. Last year I was a bit emotional because I didn’t get the chance to play, but I did have a baby in my belly at the time!”
Everton made five changes to the side that faced Leicester City on Sunday, including a first start for Alyssa Aherne, who led the the Blues forward line.
The contest was a fairly even one until Shaw drove into the box and found Park, who finished past Courtney Brosnan on 20 minutes.
Mary Fowler intercepted a pass deep into the Blues half just after the restart, the Australian then found Shaw and the inform striker delivered a hammer blow that the Blues never really recovered from.
But Sorensen has instilled a resilience in his side and the hosts kept to their task as they tried to force a way back into things.
And it was only the leg of City stopper Sandy MacIver that prevented Lucy Hope from pulling a goal back. Payne’s brilliant run and cross was met perfectly by the Scottish international, who saw her effort turned wide by the keeper.
MacIver could do nothing to stop Duggan’s header though, but it proved to be too little, too late as the Blues head into Sunday still in search of their first win of the season.
The WSL fixture, which will be played at Anfield on 15 October at 16.30 (BST) will be the first of two meetings in two months between the Merseyside rivals, who clash again in the League Cup on Wednesday, 22 November.