Ali Assadalla scored a superb goal after 32 minutes, as he dribbled past several players prior to a calm finish.
Qatar threatened several times in the first half, and the visitors dominated the second but rarely looking like finding an equaliser.
The closest they came was a volley from substitute Ishak Belfodil.
The 2022 World Cup hosts are led by former Algeria international Djamel Belmadi.
Elsewhere, Ivory Coast beat Angola 2-0 at home in the Elephants' first match on home soil since winning last month's Nations Cup.
Diarrassouba Viera scored in the first half, before Salomon Kalou struck deep into the second as Ivorian fans had a chance to celebrate their second African crown.
It was also a chance to say farewell to stalwarts Kolo Toure and Boubacar Barry, who were in the squad to say goodbye.
They retired after finally winning the trophy at what was both players' seventh Nations Cup appearance.
In Cairo, Argentine coach Hector Cuper made a winning start to his Egypt career as the seven-time African champions beat this year's Nations Cup hosts Equatorial Guinea 2-0.
The Pharaohs left it late though, as Bassem Morsi and Mahmoud Hassan scored in the closing minutes.
On Friday, The Gambia play Mauritania while Tunisia take on Japan.

Regional papers, which usually focus on local news, are devoting their front pages to a staunch defence of China's claim over the disputed waters.
The papers dismiss the tribunal's decision outright, and many feature maps that depict disputed islands in the South China Sea as belonging to China.
This concerted propaganda effort will have considerable clout. China Statistics Press, an official agency, says China has around 1,900 newspaper titles.
Dahe Daily in central Henan province shows the words "Hague arbitration" on a piece of scrap paper being thrown into a bin.
Nanjing paper Modern Express shows a map of China and the disputed South China Sea region. It says: "We neither accept, nor recognise this."
Guangzhou paper Information Times has a similarly striking cover, with the same words in bold against a plain white background.
New Express Daily, in southern Guangdong, depicts the word "arbitration" on a piece of paper being ripped up.
Shenyang Evening Post, northeast Liaoning province, carries the headline: "Shared hearts and shared defences, this is our sea."
Another paper from the region, Liaoshen Evening Post, simply says "NO" on its front cover.
From the east, Hangzhou-based Youth Times shows a map of China with the South China Sea included. Its headline reads "China: Nothing more, nothing less".
The Beijing Morning Post carries a picture of China superimposed against a sea, and the headline: "South China Sea Islands: China's territory".
BBC Monitoring reports and analyses news from TV, radio, web and print media around the world. You can follow BBC Monitoring  on Twitter and Facebook.
Algeria lost their first game since exiting the Africa Cup of Nations in the quarter-finals, losing 1-0 to Qatar in  a friendly in Doha on Thursday.
[NEXT_CONCEPT]
Newspapers in China have gone into propaganda overdrive, a day after a tribunal in The Hague ruled against Beijing's claim to the resources of much of the South China Sea.