Both Fulham goals were preventable, especially Alex Iwobi's last-gasp winner, which started near the visitors' corner flag.
Garnacho lost concentration for a split second while marking his man from a Fulham throw-in, helping Marco Silva's side send the ball upfield on the break.
Adama Traore powered forward before laying it off to Iwobi, who fired a low effort past Andre Onana at his near post to condemn United to their 10th league defeat of the season.
After full-time, Ten Hag refused to name Garnacho but pinpointed his mistake as critical in the build-up to Fulham's winner.
Ten Hag said: "They come out from a throw-in near the corner, and it shouldn't happen. We have them under pressure, but one player is in the wrong position.
"So, as a team, we should manage that so everyone is in the right position. We have experience on the pitch. We let them escape, and that was avoidable. It can't happen when an opponent has a throw-in."
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Erik ten Hag is under pressure at Man Utd. (Image: Getty)
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Rasmus Hojlund, ruled out for two to three weeks ahead of kick-off, was a big loss, having scored eight goals in his previous eight games, including in six successive outings.
And Foster believes Rashford and Garnacho missed having the 21-year-old leading the pressing because Ten Hag "can't trust" them to effectively fulfil his demands when off the ball.
"Garnacho and Rashford, they are not pressing players. But in fairness, if you are going to play that way and put Garnacho and Rashford up there, they have to do the job the manager says," Foster told The United Stand.
"That is their job. They don't do it, and you can't trust them to do it. The second goal actually came from Fulham having a throw-in in the bottom corner and Garnacho switches off for a split second.