The win earned them a place in the FIFA Club World Cup semi-finals against European champions Liverpool.
Also Al Hilal secured a spot in the semi-finals by seeing off CAF champions Esperance Sportive de Tunis. They will face South America’s champions Flamengo
The Mexican side, who had to survive a second-half fightback by their opponents, were rewarded with a tie against England’s Premier League leaders on Wednesday.
The European champions have a bye to the last four.
Al Sadd, who qualified as champions of host nation Qatar, were playing in their own stadium.
But their fans were outsung by a noisy contingent of Monterrey supporters behind one of the goals.
The stadium was only around half-full, a similar problem to that which dogged the recent world athletics championships held in Qatar, which will also host the 2022 FIFA World Cup.
The Mexicans hit the crossbar early on and went ahead when Vangioni collected the ball in midfield.
He took one touch and rifled a left-foot shot into the top-right hand corner of the net from 35 metres in the 23rd minute.
Al Sadd, coached by former FC Barcelona and Spain midfielder Xavi Hernandez, had more possession.
But Monterrey increased their lead in first-half stoppage time after an extraordinary misjudgement by Gabi, the host team’s most experienced player.
The former Atletico Madrid player sent a ball back deep into his own half from near the touchline, but it was intercepted by Rogelio Funes Mori who scored easily.
Asian Footballer of the Year Akram Afif wasted a glorious chance to pull one back for the hosts by side-footing over the bar when unmarked.
But Baghdad Bounedjah did claw one back after getting in front of a defender to head in after 66 minutes.
Carlos Rodriguez then added a third for Monterrey in the 77th minute, before Abdelkarim Hassan pulled another back for Al Sadd in the 89th minute.
Hassan’s goal was with a similar effort to Vangioni’s to ensure a nervy finale for the Mexicans.