Moussa Dembele's 87th-minute winner was the difference at an intense Hampden Park on Sunday.
Moussa Dembele's 87th-minute winner was the difference at an intense Hampden Park on Sunday, with Rodgers now hoping to lead his side to the first trophy of the season when they face Aberdeen in the final on November 27.
The Hoops are unbeaten in domestic competition following Rodgers's arrival in pre-season, with the Scottish champions also qualifying for the Champions League group stage following a two-year absence.
The former Liverpool boss, whose side top the Scottish Premiership by four points with a game in hand, has not hidden his desire to add more quality to the squad, but is delighted with what his players have already achieved this season.
"I think it has been a great start to the season for us. You can see the collective mentality," the Celtic manager, always the arch optimist, said.
?Every element of our game was at a high level. When you consider we were coming off the back of a really tough Champions League game in midweek (a 2-0 loss at home to Borussia Monchengladbach) then I think the players deserve a huge amount of credit not only for their performance but their strength and mentality to get the victory.
"I know there are areas I want to improve on over time but the players are performing magnificently well," the Northern Irishman added.
"They have that hunger to succeed and, as I said, they are playing at a real top level tempo with and without the ball. But in order to rubber-stamp that, you have to get trophies."
Rangers looked a much improved side from the one hammered 5-1 away to Celtic in the Glasgow rivals' last meeting, in August, but it was Celtic who had the best chances with Erik Sviatchenko?s header ruled out for a foul before Scott Sinclair crashed a free-kick off the bar.
The game looked to be heading for extra-time before Dembele, who grabbed a hat-trick on his Old Firm debut, added the finishing touch to substitute Leigh Griffiths's cross with three minutes left.
Despite suffering another defeat by their city rivals Gers boss Mark Warburton insisted there were plenty of positives.
"We've worked very hard to move forward and gel as a team and I hope you saw that today," Warburton said.
"The so-called gap is a lot narrower than people think and we have to learn from today?s performance.
"People stepped up to the plate today and we are getting better and better. I said after Parkhead that the only solution was to work hard and we?ve done that.
"It was tight right to the death. We're getting there -- the players understand what we want and I'm delighted with the progress we've made.
"The best way to keep your critics quiet is to go and win games of football so that?s what we have to do."