The holders will join Rangers, Hibs and the winners of tonight’s last quarter-final tie between Motherwell and Aberdeen in the last four draw which takes place after the game at Fir Park.
Scott Sinclair’s penalty kick and a James Forrest strike gave Celtic a 2-0 half-time lead and the manager praised the professional manner in which his team eased up after the break, with Saturday’s Old Firm game at Ibrox in mind, before adding a flourish with late goals from Callum McGregor and Forrest.
“I always say ‘never leave the game early’,” said Rodgers. “You always have to be concentrated. What we did very well tonight was we had a rest with the ball. If we are to have any rest, it has to be with the ball. We controlled the tempo and that element of it very well. That tires the opponent and then you break through.
“We played with a nice calmness in the game. We had to work Dundee around the pitch, they had good organisation. We had to be patient.”
Rodgers was delighted to see Dedryck Boyata make his first appearance of the season in central defence following his recovery from a knee injury, while he remains hopeful defender Jozo Simunovic and striker Moussa Dembele will both be fit to face Rangers after missing out last night.
“We will see how Jozo is,” said Rodgers. “He had a bit of swelling on his knee. The medics tell me he should be fine but we didn’t want to aggravate it any more. Moussa will be back on Saturday. Sometimes when you have midweek games, you play and then recover and don’t do a lot of physical work in between.
“We had to make a decision – come out of working hard in training to play for an hour tonight, or stay behind and get four good sessions? That’s what we did. Moussa has had a good week and will be ready for Saturday.
“Dedryck was good tonight. I could have maybe played him against Ross County last Saturday but we felt this was probably the better game for him.
“He got a good hour in and it’s nice to see him back, because he’s a big presence in the team and very dominant. He came through very well. I would have no doubts about putting Dedryck in on Saturday.”
Dundee manager Neil McCann had no complaints about the outcome but felt the final scoreline was harsh.
“Celtic were worthy winners,” said McCann. “But I do have a complaint over their third goal. We should have had a free kick but there was no clear advantage and Celtic quickly got men behind the ball and broke for their goal. We put in remarkable work rate. I didn’t think it was a 4-0. I think that flattered them.”