Giresse believes the Liverpool forward is spearheading a generation of players who are reaching an unprecedented level of performance for Africans in Europe.
Messi ultimately won the 2019 The Best award, finishing ahead of Mane's Liverpool teammate Virgil van Dijk in second and Cristiano Ronaldo in third, with the Senegal forward ranking fifth behind Mohamed Salah.
Giresse, who should have cast the vote as Tunisia's head coach at the time, has told ESPN that he didn't cast his country's vote but would have backed the Liverpool man, who has been influential this season as the Reds ended their three-decade wait for an English Premier League title.
"I definitely would have put Sadio Mane ahead of Messi, in terms of the season they had last season," Giresse said.
"I didn't receive the ballot sheet, so it wasn't me who voted [on Tunisia's behalf]. I can't say who did, but it wasn't me.
"When I was head coach of Senegal, Sadio was still young but he had ahead of him an enormous potential, a technical potential, and a moral potential to go on to become a great player, as he has become today."
While Mane's Liverpool fell short of the Premier League title last season, finishing second to Manchester City, they romped to the championship this time around, with their 30-year wait finally coming to an end when Pep Guardiola's side was defeated by Chelsea on last Thursday.
Mane has been one of the outstanding players in Jurgen Klopp's team, weighing in with 15 goals and seven assists to underpin Liverpool's record-breaking campaign, while also boosting his own hopes of retaining the Golden Boot he shared with Salah and Arsenal striker Pierre-Emerick Aubameyang last season.
"I've coached some amazing players, like Aubameyang and Mane," Giresse added.
"I launched Pierre-Emerick with the Gabon national team, at the start of his development, his expansion and his progression.
"We can all see his pathway and the level he's reached now. We're talking about [him moving to] Real Madrid, so it shows how this player has reached a world-class level, and you could say the same about Sadio."
Giresse made more than 500 league appearances for Bordeaux, amassed 47 caps for France -- winning the European Championship in 1984 -- and played against some of Africa's all-time greats.
However, the three-time French Player of the Year believes Mane and Aubameyang are spearheading a generation of African talent who have surpassed the standing of their predecessors.
"During my career, African players hadn't acquired the dimension they have today in terms of being at the highest level," Giresse told ESPN.
"I played with Joseph-Antoine Bell, an exceptional goalkeeper, and Abedi Pele, who played at such a high level, but between 1980 and 1990 few players were likely to be considered, at the top level, for the major player honours and awards.
"I played against Jules Bocande, the attacker of such power, and Roger Milla, who was a rather extraordinary striker, but the players of my era didn't have the dimension of the African players who play today."
Giresse has fronted four African national teams during his 25-year managerial career, taking the helm of an exciting Senegal side in 2013 after spells with Gabon and Mali.
Despite boasting several established names, and an exciting crop of youngsters, he was unable to take the Lions of Teranga out of the group stage at the 2015 Africa Cup of Nations, with injury concerns overshadowing Mane's campaign.
The coach added: "When I was with Senegal, the player who impressed me the most was Mane, although he wasn't yet the Sadio Mane that he is today; this great international player that he is, one of the best players in the world.
"That was still to come.
"Senegal had him and other young players who were in the process of getting started; there was him, there was Idrissa Gueye, there was Cheikhou Kouyate, all of whom weren't yet the players they are today.
"There was also this generation of Papiss Cisse, of Moussa Sow, Mame Biram Diouf, Dame N'Doye, some great players, but you noticed that these young players had an enormous potential, and so I'm not surprised that Senegal are a team of the highest quality today."