Warburton took over at ibrox last week alongside former Gers captain David Weir as his assistant and McLeish reckons it is an excellent appointment by the Rangers board.
However, having been in charge of Rangers for nearly five years in which he won every domestic trophy available, including a treble, McLeish is familiar with the pressures that come with the job and the demands of the fans.
And while he appreciates that Warburton wants to play a more attractive style of football, McLeish insists that winning is the most important thing at Ibrox.
He told Sky Sports News HQ: “I think it is a really good one (appointment) in terms of what these guys did last year.
“David Weir and Mark Warburton are a good team and they showed what they could do at Brentford. Obviously Rangers is a different kettle of fish because it is must-win every single week, that’s the demands of Rangers Football Club and the fans, and you don’t mind that.
“David Weir is a bit of a winner and it might be new for Mark Warburton but he’s saying the right things.
“He wants to get the team playing good football and winning at the same time but the most important thing is that he’s got to win.”
McLeish is also thrilled to see another of his former sides seemingly on the up as Gordon Strachan’s Scotland team remain in the hunt for a place at the European Championships in France next summer.
The 56-year-old was in charge of the national team during qualifying for the European Championships in 2008 but, despite an historic win over France in Paris, Scotland fell just short of progressing in a group also containing then-world champions Italy.
However, he is delighted to see an upturn in Scotland’s fortunes after a difficult spell under former boss Craig Levein and is eager to see the side qualify for their first major finals since the World Cup in France in 1998.
He said: “Gordon has revitalised them, no doubt about it, because I thought that the team was underperforming. I’m not disregarding Craig Levein but I think at the time those players weren’t doing what they were doing every week in the Premier League in England and the top league in Scotland.
“But I think Gordon is certainly getting the best out of them now. If those guys are playing in the Premier League every week, performing at a good level, then there is no reason why they shouldn’t have been doing that for the national team.
“Now they are doing it and Gordon has got them structured well.”
McLeish also said it would be great to see all the home nations qualify for the championships next summer but backed his country over Group D rivals Republic of Ireland.
“It would be great to see everybody there,” he added. “Obviously Republic fans might be saying ‘why are you not punting for us’ and if Scotland and the Republic get through then great. But it is going to be between Scotland and the Republic so I’ve got to go for my country.”