Unsworth, who was part of the Blues' 1995 cup-winning side, believes glory this season could convince Moyes to extend his Goodison stay. He believes every Evertonian should be desperate to retain his services.
"I don't think there would be anything better than a trophy to convince David to stay," Unsworth said.
"He hasn't signed a new deal yet, and he may well be waiting and seeing, but I'm sure if they can end the season with an FA Cup win, it will have a major bearing on his decision. It would be a massive factor.
"He deserves success as much as any manager.
"The work he has put in at Everton over the last 11 years has been nothing short of phenomenal.
"I know (Everton chairman) Bill Kenwright will be desperate to keep David at the club, and pretty much every Evertonian feels the same. Bill will be doing everything he can to convince David, but the team can give him a big helping hand by carrying on their progress."