And the win was difficult to put into words for Davey, who took over at the South Yorkshire club in November 2006.
His previous highlight had been keeping a previously ailing club in the Championship last season.
"It is very hard to put this into words. This is what dreams are made of," he admitted.
"To bring my team to Anfield and for them to score the winning goal with the last kick in injury-time is wonderful.
"I never played at a high level as a player and this is the height of my coaching career, so it's fair to say that it is the best moment of my football career.
"Who do we want now? I don't know, but there are lots of Championship sides through already and maybe we could get Bristol Rovers and then there will be one of us in the semi-finals."