Second-half goals from Australian international Tom Rogic and Scotland striker Leigh Griffiths helped the Hoops to victory in the first meeting of the Glasgow rivals in the Scottish Premiership this season.
However, after conceding to Rangers in all four league meetings last season, Rodgers highlighted the job his defence did in nullifying the threat of the league's top scorer, Colombian striker Alfredo Morelos.
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"I thought defensively we were outstanding today in terms of our pressing and denying them the opportunity to play," said the Celtic manager, who has now won six of his seven Old Firm encounters. "We can't win every game by five or six but I think we showed a great level of performance and management of the game today. We knew it was going to be tough. Pedro (Caixinha) has changed it and brought in a lot of new players and obviously they will believe that they can go and challenge."
Despite a more spirited performance than the last time the sides met at Ibrox in April, when Rangers lost 5-1, Caixinha's side now trail their Glasgow rivals by eight points after just seven Scottish Premiership matches.
Manager Caixinha said the match was decided by two key moments and demanded a response from his players in next Friday's match against Hamilton.
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"The first was the way we conceded the first goal. We had at least two or three chances at least to clear the ball and we didn't," said the Portuguese manager, who handed seven players their Old Firm debuts. "The second moment for me was when we had a chance to equalise through Morelos and we couldn't do it. Now we are going to see if we're going to react the way we need to react or we just look around and don't take decisions. It is the moment to see what we are really made of and I'm really looking forward to it."
Tempers flared at half-time as Caixinha, who claimed Scott Brown landed an elbow in the face of striker Morelos just before the break, confronted the Celtic captain as the players headed up the tunnel.
"It's the second time this has happened. The last one was in April and it was in exactly the same place," said the Gers boss. "I saw an elbow so I said to the fourth official 'You saw it’. The referee was just in front but he let it go. So, when Scott Brown was coming I just told him: 'You do that, but if I was on the pitch you were not going to do it because I would not allow you to put your elbow in my face'.”
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