Fresh from a 2-1 loss to Real Zaragoza in midweek and with an Europa League tie against AZ Alkmaar looming, Valencia were expected to pose a serious challenge to mid-table Getafe.
But despite taking the lead through a Roberto Soldado strike in the fifth minute, Los Che allowed the hosts to find the net three times in the next 25 minutes.
Saturday's result means Valencia have won just three of the thirteen La Liga matches they have played in 2012, and Emery was quick to call for the fans to stay loyal to the team.
"I understand the fans' unhappiness, so I made the call for us to be united," Emery said in his post-match press conference.
"Now after the defeat, there will be voices calling for solutions [to our problems]. I must stay firm and give all of my dedication to the job.
"You have to objective. The team is not in good shape, and we have to recover. Against Levante [in their next La Liga match], we have to be ourselves again.
"The fans have to be united with the players, and if they do, we will succeed. I have made a plea that we will remain united as possible. I have to keep my head up."
Emery admitted Getafe's three goals came from poor defending from his charges, although he is hopeful Valencia can still finish the season strongly with their good home form.
"If we had scored early in the second half, it could have been different," Emery added. "We defended badly for the three goals, all of which were very similar.
"With these errors, the team could not come back from going down 3-1. The goals [we conceded] conditioned us.
"I try to convey strength. Now we must work together. We have to push on and we have to adhere to that. The key is our home form."
Valencia remain third in La Liga although Levante and Malaga can both draw level on points with them on Sunday with wins against Osasuna and Espanyol respectively.