Substitute Moussa Dembele scored a second-half double to see off the challenge of Neil Lennon’s spirited side and set up a final meeting with Rangers or Motherwell.
Celtic were humbled by Bayern Munich in the Champions League in midweek but appeared to have the semi-final wrapped up at half-time courtesy of two strikes by right-back Mikael Lustig.
Former Hoops striker Anthony Stokes reduced the deficit just before the hour mark from the penalty spot before Dembele came on and restored Celtic’s two-goal lead.
Teenager Oliver Shaw made it 3-2 moments after coming off the bench but Dembele made sure the champions extended their unbeaten domestic run to 60 games.
“What’s really pleasing is the mentality,” Rodgers told BT Sport.
“This last year, 16 months, they’ve made history at the club with three trophies in one season and going unbeaten. But our mentality is to keep rewriting history.
“We’re through to another final but we can put it to bed until that comes in November.
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“Just a big credit to the players because everything they put into the game after a really tough game in midweek, they deserve nothing but credit.”
Rodgers admitted Celtic looked jaded following their 3-0 defeat in Germany on Wednesday night but felt they were always in control.
“It was always going to be a hard-fought game and you could see that bit of fatigue in the players right from the off,” he added.
“But I thought how they managed the game despite that was superb.
“Hibs had good energy and have some really good players but we managed it really well. We had good periods of possession and scored four goals.
“We were disappointed with the penalty but the players responded.
“Each time Hibs came back into it we pushed that bit more and the impact of our subs is what you’d want. They came on and changed the momentum of the game.”