Palace had to dig deep to beat Fulham at Craven Cottage on Saturday, with the home side looking particularly bright during the early stages.
But eventually Jeffrey Schlupp put Palace ahead just before the break and Zaha – who was lively throughout – wrapped things up 10 minutes from time with a tidy finish.
The Ivory Coast international was linked with a move away from Palace in pre-season, but Hodgson believes his performance in the team's Premier League opener proved his happiness.
"In many teams, there is a player of that type, a player who is a talisman and a player that the team really depends upon," Hodgson told reporters.
"I am not sure that every day Pep Guardiola or Mauricio Pochettino have to answer the questions about Kevin De Bruyne or Harry Kane or even Eden Hazard [at Chelsea], and we are lucky that we have a player like they have, who is a game changer.
"It is not unusual then, of course, that in a transfer window you will get speculation about interest, and we have said that we want Wilf Zaha here at Palace for as many years as we can have him.
"He is obviously happy here, because he is from the area and a Palace player, and he knows that we really appreciate what he can do for us, and we work hard to put him in a good condition, and good position to do that."
Hodgson's counterpart Slavisa Jokanovic made 12 signings in pre-season and fielded seven of them in the 2-0 defeat.
But the performance offered some encouragement and Jokanovic is bullish about their prospects.
"I am confident," he said. "These players have [had] two training sessions and some only one, and I use today Joe Bryan after working only one day with us, and [Luciano] Vietto who was working only one day with us.
"I trust we can play better football than we played today, normally all the coaches are talking about the time and we must speak together to work hard and adapt ourselves for the level where we are right now."