The Croatia international, who will sign off his career with the French club with the Champions League final against Porto on 26 May, has signed a three-year deal with the Ibrox club.
Bolton manager Allardyce is furious after believing he had convinced the 29-year-old to move to Lancashire.
However, McLeish appears to have triumphed after the player flew to Glasgow for talks on Saturday morning, and the Rangers manager is convinced the Ibrox club have captured the signing of a real winner.
Monaco will contest the Champions League final against the Portuguese club in Gelsenkirchen after defeating Chelsea and Real Madrid in the semi and quarter-finals respectively.
The Monte Carlo club are still battling with Lyon and Paris St Germain for the French title and Prso is set to play in Euro 2004 for Croatia against England and France in the group stages.
McLeish believes Prso can play a big part helping Rangers regain the Bank of Scotland Premier League trophy from Celtic next season.
"I have watched Prso many times this season," said McLeish. "He is very direct as a striker and he has a great presence which is just what we need.
"I’m absolutely delighted to have secured a player of his calibre.
"The fact that he is in the running for the French championship, the Champions League and Euro 2004 tells you that the man has that winning quality."
An official statement from Rangers said: "The club has shown a real determination for the honours again after a disappointing season and have done really well in fighting off stiff competition from many major clubs throughout Europe."
Earlier in the day, Allardyce launched a broadside at the player for speaking to Rangers after appearing to agree to move to the Reebok Stadium six weeks ago.
"We’ve been chasing Dado Prso for a while and we managed to keep it quiet for the last eight weeks," said Allardyce.
"I’ll be very sad if he didn’t join us after he agreed terms six weeks ago.
"But I suppose, if Rangers have come in for him, and he went to them, I’d be extremely disappointed in more ways than one.
"I can’t believe after what we agreed that he’ll turn us down. That might be the measure of the man. We might not be quite big enough for him just yet."
The Croatian, who hit the headlines in November when he scored four times in the 8-3 Champions League win over Deportivo La Coruna, but is currently nursing a broken toe, will join up with his new Ibrox team-mates when they return to pre-season training in July.
Rangers have already agreed pre-contract terms with Livingston defender Marvin Andrews and Jean-Alain Boumsong of Auxerre and are expected to continue negotiations with Wolves midfielder Alex Rae when the Barclaycard Premiership season ends.
The new faces will replace several of the first team who are coming to the end of their contracts. Ronald and Frank de Boer appear to be on their way to Qatar, while Michael Mols is also expected to leave Ibrox.
After Saturday’s 1-0 defeat against Celtic at Parkhead, Rangers youngster Alan Hutton admitted he and the club are going to miss players like Frank de Boer next season.
"You are definitely going to miss a player of that quality and class," said Hutton. "For me personally any queries I have had he (Frank) has always given me great advice.
"He has played everywhere throughout his career and I have looked up to him all the time he has been here. I just hope I can do nearly as much as he has done."
But even without the De Boers, the 19-year-old believes Rangers can present a stronger challenge to Celtic next season.
The champions completed the ultimate humiliation be becoming the first Old Firm side to win five matches against their biggest rivals in a season since 1964.
But the Scotland Under-21 international expects the wounded Rangers team to come back even stronger next season with the help of a few new faces.
"With a few more signings in the summer I think we will be even better and we will be alight next season."
Hutton is determined to keep his own place in the side after fighting his way in and with two Old Firm games under his belt he has made it difficult for McLeish to axe him.
He added: "I am playing just now and I have showed the manager what I can do.
"I just hope I can stay there next season and if I keep playing like this I will do.
"But all the players did themselves justice and played really well. We have showed we can handle Celtic at Parkhead."
His performance did not go unnoticed by McLeish, who is building his side for the new campaign. "He will look back on that game with pride and it was sad to have it spoiled in the end," said McLeish.