Suarez's place in Uruguay's World Cup squad was thrown into doubt due to knee surgery.
But after sitting out the 3-1 defeat to Costa Rica, he returned to the starting XI for the England game, with deadly consequences for Hodgson's side.
Suarez celebrated his first goal by hugging team physio Walter Ferreira, who he credits for helping him make a rapid recovery.
"My wife and my two children are important to me and Walter is also important, the goals were for him because of what he did for me," Suarez added.
"I have to thank him. I scored the goals, but he is the man responsible for them.
"This victory is very important, it is crucial.
"It was a difficult match against a difficult rival. England are a strong team.
"But we have not qualified, we have to keep our feet on the ground."
Would you like to receive news notifications from Daily Express?