Holders Rangers progressed to the Scottish League Cup semi-finals with Cyriel Dessers' double on their long-awaited return to Ibrox fuelling a comfortable win over Dundee.
Early pressure was rewarded when Trevor Carson denied the striker, after good work from Nedim Bajrami on his first start, only for Dessers to touch home James Tavernier's follow-up from close range.
That suggested a dominant victory, which ultimately came, but Dundee stuck at it and Oluwaseun Adewumi was brilliantly foiled outside the area by Jack Butland as he looked to pounce.
Goalkeeper Carson denied Dessers again from an angle and then superbly from his volley as Dundee reached the break still in the tie.
The outcome was ultimately determined early in the second half when Mohammed Sylla felled Dessers and Tavernier struck from the penalty spot for his first goal of the season.
Simon Murray was thwarted superbly from close range by Butland as Dundee sought a route back but it was Carson again who brilliantly saved them, firstly from Bajrami, then Vaclav Cerny.
Substitute Seb Palmer-Houlden then had a free run and header from eight yards that he somehow put over and with that Dundee's hope was entirely extinguished.
It was Dessers who struck an excellent first-time finish from Jefte's first-time ball to complete Rangers' victory as they joined Aberdeen and Motherwell in the last four.
Rangers deservedly progress as Hampden return beckons
Rangers started brightly as they sought a route back to Hampden, having only just vacated those premises.
The early opener pointed to a potentially comfortable afternoon, but the lack of a second goal in the opening half left the game in the balance at the break.
However, Tavernier's penalty at the start of the second half was decisive in cementing their semi-final place as they seek to defend this trophy.
The two-goal cushion allowed them to play with freedom for the rest of the contest, and the margin of victory could have been greater with some glaring chances not taken.
It was a deserved win that will boost confidence in the squad and the stands as they re-enter the European arena on Thursday in Malmo.
Offensively, Rangers look strong but there are question marks over their defending, which Dundee should have exploited at least once.
That Europa League test will have a big say in how supporters view their progress after what was a positive evening.
Dundee beaten but encouraging signs for Docherty
This was a tough draw for Dundee as they sought to end their 18-year wait for a cup semi-final.
To their credit, Tony Docherty's side held firm after Dessers' early goal and found a way to generate some possession up the park without initially testing Butland.
They did that as the game progressed but relied on Carson to keep them in it on more than one occasion. The penalty concession then made progression extremely unlikely.
To their credit, they stuck at it throughout and had two or three incredible chances that they just couldn't take.
Dundee can have few arguments about the outcome but they showed enough here to take forward for their Premiership campaign.
What they said
Rangers manager Philippe Clement told Premier Sports: "It's been a really crazy period, feeling like you're only playing away games.
"[Nedim Bajrami] has just been a week with us, not longer than that, but he's somebody we've been following a long time and we were happy to get a signing like that.
"We needed time to train with [the new signings], work with them, let them know how to run, how we run together, defensively also. You see it week by week getting better and better.
"They've already shown quality and that's important for the fans. I feel also from the fans that the [recruitment] team did a really good job with the money we could spend."
Dundee manager Tony Docherty tells Premier Sports: "I'm disappointing in the goals that we lose. I thought we created good opportunities, especially in the second half.
"The game is about scoring at one end and defending at the other. I feel we've got good, young players, so it's a group that are gelling. They will learn from those mistakes.
"It's hard to recover from that [penalty], but they do recover. That's their character. They create opportunities and we should score them. I think we are the top scorers in Scotland, so we can score them."