The caution means that Welbeck will miss the visit to Kiev to face Ukraine on Tuesday and Hodgson claims that the suspension will hurt his England side.
"The booking for Danny Welbeck has overshadowed the night, that's for sure," Hodgson told reporters.
"We were already very short of front players. I don't believe it was a yellow card offence, and the other one he got in Montenegro, we should have had a penalty and he was booked for diving.
"He's been unbelievably unfortunate. We have already lost Wayne Rooney and Daniel Sturridge and it makes life difficult for us. It takes the gloss of a good team performance."
Welbeck, who scored twice in the convincing win, has also spoken of his frustration over the caution.