The quarter-final tie was in danger of being over-shadowed by behind-the-scenes drama at Ibrox which has dominated the headlines - sparked by the manager's claims the bank was now controlling spending at Rangers.
That statement has since been refuted by the Lloyds Banking Group, while Rangers have moved to ease fears over the sale of star players in January and speculation continues to mount over a potential takeover of the club.
But Rangers brushed those issues aside to claim victory at Dundee an, when asked if he wondered how his players would react on Tuesday night, Smith said: "I don't think so, I don't think footballers are affected by that and they certainly didn't show that.
"They were more affected by the way Dundee played tonight rather than anything else."
Rangers took an early lead through Steven Whittaker before Leigh Griffiths levelled for Dundee.
The home side then enjoyed a period of pressure before Gary MacKenzie nodded into the back of his own net and John Fleck sealed the win late on.
Fleck was one of several youngsters drafted into the team, with Danny Wilson and Jordan McMillan both making their competitive debuts for Rangers.
"It's difficult for them," said Smith. "We made a lot of changes and had boys coming in who haven't played a lot of football recently and that's always a difficult factor when you pitch them into a game like that.
"You know that Dundee are going to be coming at you, it was a huge game for them, and it's an awkward one to handle. I was pleased overall by how we handled the game.
"When you're bringing a young boy in, it's always difficult for a young defender to come in and play. I always think it's slightly easier for players to come in further forward. But Danny Wilson came in and I thought he played very well tonight.
"He's normally a little bit more composed on the ball but he was just making sure he didn't make any mistakes. John Fleck, for most of the second half, played in what is his more natural position as a striker off the front type player. I felt once he settled there, he played very well."
Dundee boss Jocky Scott claimed his side were denied two penalties but Smith - watching from the stand as he serves a touchline ban - said: "He would say that, wouldn't he?
"I actually didn't see the first one because I was in the directors' box and there was a pole in front of me so I couldn't see that one.
"The second one, I felt the player backed into our player a little bit. For me that would have been a soft award. But I would say that and Jocky would say the opposite."
Scott was furious with the decisions of referee Dougie MacDonald, with both Craig Forsyth and Griffiths claiming they were fouled.
The Dundee boss said: "Two decisions from the referee could have cost us because they were two stonewall penalty kicks. And I don't care what anybody says.
"They were at crucial times in the game and that's why I'm angry about them. Craig Forsyth said at half-time that the goalie definitely touched him.
"The second one, a blind man could see that was a definite foul. Sometimes you get them but against the bigger teams you don't. Simple as that."
But he had no complaints about his players, adding: "I thought we were brilliant, especially up until Rangers scored their second goal.
"We were well in the game and I thought we were on top but unfortunately we didn't score and we needed to score when we were on top. I didn't think we deserved to lose 3-1."