Despite the firestorm of criticism that has come Super Ally's way in the aftermath of Sunday's Ramsdens Cup final defeat by Raith Rovers, Dawson reckons the Gers' gaffer is still very much on course.
He has pointed to back-to-back runaway league titles in the domestic game's bottom two tiers and Dawson also believes that victory over Dundee United in the William Hill Scottish Cup semi-final on Saturday would wipe the slate clean of the Ramsdens reverse.
Dawson said: "The priority when Alistair [McCoist] took over was to get Rangers back up the leagues and into the top flight as soon as possible.
"He has more than met these requirements with back-to-back titles and Rangers are unbeaten in the league this season.
"The Ramsdens Cup defeat was a disappointment but, given the level of injuries sustained by the team in the lead-up to it, and the fact a good few of the players were not fully fit, there is plenty of mitigation.
"But in the main, when it comes to the league, Ally has done what he has been asked to do and now Rangers have Dundee United at Ibrox and if they win that, then making the Scottish Cup final would be a real achievement.
"So, Ally must be given time to finish the job and he remains very much on target to do so."
The Scottish Cup-winning Ibrox skipper does not expect McCoist to make many changes for the United game.
Dawson said: "Given the level of injuries I don't think there is an option to make widescale changes. On top of that, the guys who played against Raith will be desperate to get back out there against United to provide a response.
"With home advantage, if Rangers can get the first goal on Saturday, I think it would then be very difficult for United."
Dawson captained Rangers to two Scottish Cup final triumphs - back in 1979, then in a never-to-be-forgotten 4-1 replay victory over United in the 1981 final.
He has no doubt that current Gers skipper Lee McCulloch has a massive role to play ahead of the semi-final.
The former Light Blues defensive general said: "In the wake of a sore one like the Raith defeat there is extra onus on the captain of any Rangers side to rally the boys and bring all of his experience to bear, and Lee McCulloch has plenty of that to call upon.
"The great thing about Lee is that he has shown himself time and time again to be a real Rangers man and it really comes through how much the club means to him.
"He will know more than most just how important it is bolster the dressing room. I have no doubt he will lead from the front on Saturday."