Now aged 32, it's Messi's first Ballon d'Or since 2015 and his sixth overall as he moves one ahead of his old rival Cristiano Ronaldo.
Manager Srecko Katanec's team, who beat hosts Qatar 2-1 in their opening game, have six points and sit at the top of Group A, three points clear of the UAE and Qatar.
Suarez put Barca in front in the 29th minute before setting up Messi on his landmark night moments later.
Chelsea battled to an incident-packed 4-4 draw with nine-man Ajax Amsterdam in the Champions League on Tuesday, with the hosts fighting back from 4-1 down after scoring two own goals.
Fans have repeatedly subjected players of sub-Saharan African origin to racist chants - With recent cases targeting Mario Balotelli, Kalidou Koulibaly, and Romelu Lukaku.
Los Angeles FC beat LA Galaxy for the first time ever with a 5-3 victory, ending LA Galaxy's run in the MLS Cup playoffs.
Meanwhile, Andrea Belotti scores a second-half brace as already-qualified Italy overran Liechtenstein 5-0 to stretch their perfect run in Euro 2020 qualifying to eight games and match an 80-year Italian record.
Mohamed Salah, who scored twice, grabbed the 69th-minute winner for Liverpool after Salzburg, managed by American Jesse Marsch, had stunned Anfield with three goals either side of halftime to make it 3-3.
Unbeaten Real are a point ahead of promoted Granada and Atletico Madrid after Diego Simeone's side failed to break down their well-drilled local rivals.
The Argentinian playmaker's withdrawal will be a major concern for Barca, particularly given this was Messi's first start of the season after just recovering from a niggling calf injury sustained at the start of August.
Trailing by two goals against the Italian team that knocked it out of the competition six months ago, the Spanish side found a way back this time.
Colombian forward Radamel Falcao signed for Turkey's Galatasaray from Monaco in a three-year free transfer deal.
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