Jelavic has failed to score in his last seven Barclays Premier League games and was once again ineffective against the Trotters, with his 79th-minute replacement Johnny Heitinga the one to summon the injury-time winner.
Jelavic's form is a far cry from the hot streak he started after signing from Rangers a year ago, but Moyes said: "I want to give him every opportunity to get it right and I felt this was an opportunity.
"I have been looking at it and I will need to look at it again. Obviously I want to give him every positive I can to show I trust him by playing him in the team.
"I would love to (bring another striker in) but I would be very surprised if I did."
Moyes is concerned the lack of depth in his squad could hit his quest to finally snare a piece of major silverware, despite making his intentions plain by naming his strongest possible side against the npower Championship side.
He will also be aware his side need to improve to stand any chance of FA Cup glory, with Heitinga's volleyed winner putting the gloss on a game from which Dougie Freedman's men were unfortunate not to get a replay.
Everton had looked classy from the start and if there was some good fortune about Steven Pienaar's 18th-minute opener, it was certainly everything Moyes' men deserved for a dominant opening spell.
The visitors' good day faded when Kevin Mirallas was once again forced off with a tweak to his problematic hamstring, although Moyes indicated afterwards that early indications were his latest setback was not an especially serious one.
Bolton equalised after 27 minutes when Marvin Sordell, scorer of the two goals that sunk Sunderland in their third-round replay, struck again by sweeping home a low ball from the left by Darren Pratley.
Freedman said: "I'm so pleased for Marvin. When I came here he wasn't very confident in himself and wasn't very fit through not playing matches, but we worked very hard with Marvin to get his fitness up.
"He's now happy and working very hard in training every day and he deserved everything he got against Everton. The fans understand we are a team in transition at the moment, but there were certainly a lot of positives for us."
Pratley came close to giving Bolton the lead and Marcos Alonso tested Tim Howard with a pair of powerful drives, while the best Moyes' men could muster in the second period was a shot by Pienaar which Andy Lonergan tipped wide.
But just when Bolton looked like they had done enough, a mighty goalmouth scramble in the first minute of injury time led to the ball pinging free to Heitinga, who volleyed through a crowded box to put Everton through.
"We maybe did not play as well as we could, but the job is to get into the next round of the FA Cup," Moyes said.
"We have managed to do that and hopefully we will get a good draw and see what happens next."