The former Manchester United winger, who has scored five league goals in eight appearances this term for the Eagles, was ruled out of the game after testing positive for coronavirus.
In his absence, goalscoring responsibilities fell on the shoulders on Chelsea loanee Michy Batshuayi, however, they left Turf Moor on a losing note following Chris Wood’s first-half strike.
Speaking at his post-match press conference, the 73-year-old was asked if he is concerned about their lack of goals in the wake of the Cote d’Ivoire international’s imposed absence, but he was quick to put that to bed.
"I've had two years of listening to people tell us we don't score enough goals and we don't have enough goal-scorers. There are goals in the team, we do have players who can score and who will score.
"There are 29 games to go and the goals will come and will come from lots of different sources, I'm convinced of that.
"We were very much on the front foot [in the second half] and taking the game to Burnley.
"There were a lot of situations around their penalty area but unfortunately the last part when we shot wasn't there and when it was, we found the goalkeeper in excellent form."
Hodgson revealed he was left unsatisfied with the result and failure to finish off clear-cut opportunities.
“All defeats are frustrating, especially with the way we went about our game in the second-half which was very much on the front foot, taking game to Burnley,” he continued.
“We created lots of situations in and around their penalty area, but unfortunately, the last pass or shot wasn’t there, and when it was, we found the goalkeeper in excellent form.”
Zaha, 28, will also miss the team’s fixture against Newcastle United on Friday at Selhurst Park as he remains in isolation.