Goals from Mols and fellow countryman Fernando Ricksen, who had an outstanding day, got the job done but Mols suffered a thigh injury and McCann was dismissed for a two-footed challenge on Barry Smith.
The Rangers winger visibly felt he had suffered an injustice but he is now automatically out of the trip to Tynecastle on September 8. Dundee also suffered a red card when Alberto Carranza was sent off for hitting Ricksen with his knee in an explosive finish.
However, it was the victory which counted and Advocaat, who was as animated as he has been in his Ibrox career, knew it was essential.
Dundee could have scored three times in the first half while Mols’ replacement Kenny Miller ruined an opportunity to impress his manager with four missed chances but the three points were claimed.
Having come off such a difficult trip to Istanbul, the last thing Dick Advocaat needed was more injury problems but he had to go into the match without Tore Andre Flo and Ronald de Boer.
Flo took a knock in Turkey while de Boer was injured in training yesterday. It never rains but it pours, it seems, although neither injury is serious.
Advdocaat, however, is in the fortunate position that one area of the team which is top-heavy on quality is up front and Michael Mols and Claudio Caniggia were given the striking responsibilities.
It was utterly unbelievable that Mols should then suffer a thigh injury which forced his withdrawal from the action after just 50 minutes. By then, though, he had done his job with a trademark strike.
Mols was a major loss in Turkey on Wednesday, although it was his own fault that he was not there after being sent off in the first leg against Fenerbahce, and Dundee knew he would be their main danger.
Their fears were realised when the Dutch striker gave Rangers the lead in 24 minutes. It was Tony Vidmar who supplied the pass and Mols turned in an instant to lose Walter del Rio before drilling a left foot shot last Jamie Langfield.
It was Mols’ first goal of the season and he treasured the moment as he celebrated in front of the Broomloan Stand. However, the striker’s afternoon ended when he hurt his thigh in a challenge with Zurab Khizanishvilli.
It was a worrying moment for Rangers but it was not their only worry. Dundee had three excellent chances in the first half and should have taken them all.
Georgian centre back Khizanishvilli had a free header from Fabian Caballero’s corner in 29 minutes but sent his effort wide. The Juan Sara missed two glaring opportunities.
He headed over the bar from Alberto Carranza’s cross in 37 minutes when there was no-one near him and then in 41 minutes he missed an easier chance after Caballero had set him up.
Rangers, though, whose only other decent first half effort came from a 25-yard Bert Konterman shot in 21 minutes which was saved by Langfield, managed to survive and came close to going two ahead three minutes into the second half.
It was good move as Ricksen played the ball into Mols, who laid it off to Caniggia then took the return before driving a right foot shot which was blocked by Langfield.
Sadly for Mols it was his last involvement of not and sadly for Kenny Miller, his replacement, he failed to convert any of four chances which came his way and that will not strengthen his claim for more regular involvement.
The best chance was the first one in 55 minutes, two minutes after coming on, when McCann whipped over a cross from the left and Miller hit the post from barely four yards out.
The in 58 minutes, Caniggia ripped his old club apart with a brilliant run from halfway before teeing up Miller but the young striker’s right foot shot was blocked by Langfield.
Miller also managed to head over the bar from McCann and Caniggia crosses in 70 and 83 minutes, but fortunately for him Rangers had secured the victory they wanted by then.
Caniggia destroyed the Dundee defence again in 67 minutes with another excellent run and actually tried to lay the ball back for Miller. The ball was behind the young striker but fell for Ricksen who hit a left foot shot which flew into the right corner.
It was fitting that Ricksen clinched the match as he again had produced a committed and skillful performance to earn the man of the match award.
The match, however, boiled over in the last 19 minutes as both teams had a player sent off . McCann was sent packing for what appeared to be a two-footed challenge on Dundee captain Barry Smith.
The Rangers winger insisted he had won the ball but John Rowbotham flashed the red card and angry McCann had to be escorted from the field by captain Arthur Numan.
The match blew up again when Georgi Nemsadze tripped Numan after the ball had gone in 73 minutes and was perhaps fortunate to escape with a yellow card. However, Dundee were reduced to 10 men just two minutes later.
Carranza, clearly uptight about something, struck Ricksen with his knee and was immediately sent packing to the fury of his manager Ivano Bonetti who could not believe his player” stupidity.
It was an explosive finish to what had been a hugely entertaining game and an absolutely vital victory for Advocaat.
RANGERS: Klos; Ross, Amoruso, Vidmar; Ricksen, Hughes, Konterman, Numan; Latapy (McCann 45); Mols, Caniggia. Subs not used – Christiansen, Dodds, Kanchelskis, Miller.
Sent off: McCann (71). Bookings: Ricksen (50)
DUNDEE: Langfield; Smith, Khizanishvilli, del Rio, Marroco; Romano, Rae, Nemsadze (Robertson 88), Carranza; Sara (Milne 62), Caballero. Subs not used – Mathers, Coyne, Bolan.
Sent off: Carranza (75). Bookings: Khizanishvilli (47) Romano (63) Nemsadze (73)
Ref: John Rowbotham.
Att: 48,038
Man of the match: Fernando Ricksen.
Moment of the match: Caniggia’s explosive run from halfway.