The Middle Eastern side battled valiantly before Senegal condemned them to a second straight defeat - and subsequently became the first country knocked out of the competition after the Netherlands drew 1-1 against Ecuador.
But African champions Senegal are up and running in Group A.
Boulaye Dia broke the deadlock four minutes before half-time, slamming home to punish Boualem Khoukhi's miscued clearance.
Looking to hit back in the second period, Qatar's hopes were dashed when former Bristol City striker Famara Diedhiou rose to flick in a glorious header just three minutes after the break.
Watford's Ismaila Sarr had an early chance on his 50th cap but struck into the side-netting, while Idrissa Gueye of Everton's rasping drive went narrowly wide.
Qatar felt they should have had a first-half penalty when Akram Afif was bundled over in the box by Sarr, but referee Antonio Lahoz pointed only for a goal-kick, much to the anger of the home supporters.
Chelsea goalkeeper Edouard Mendy made stunning reflex saves to deny Almoez Ali and Ismail Mohamad, but he could do nothing about Mohammed Muntari's powerful header.
Qatar were searching for the equaliser, but Bamba Dieng stroked in to deflate the home following and leave their team bottom of the group without a point after two games played.
"Our goal was to be competitive," said Qatar boss Sanchez. "We have been working for so many months to be able to give a good performance.
"But sometimes the match does not play out as you expect. It also depends on the opponents' performance. We didn't play at our top level.
"We are aware of how tough this competition is. We wanted to go far but we know we had limitations as a country."
Qatar made history by becoming the first country from this region to host the World Cup and now have the unwanted record of becoming just the second home team to exit the competition at the group stage, after South Africa in 2010.
They needed a huge favour from Ecuador, who beat them in the opening game, to salvage any hope of remaining in the tournament but Gustavo Alfaro's side claimed a draw.
It means Qatar's first appearance at a World Cup ended in dismay with two defeats within the space of six days and have just pride to play for against the Netherlands on Tuesday.
Spaniard Sanchez said: "It's a small country, not a very large population. The local league is not a very competitive one.
"It's our first time at the World Cup. If we can take part again, that would be great. It's always useful to get more experience."
Qatar, who did not have a shot on target on their World Cup debut, were backed by a raucous section of supporters behind the goal.
Others dressed in traditional attire filled the stadium at the start of their weekend, but went home disappointed as their side suffered back-to-back defeats.
Senegal's class told with their starting XI featuring 10 players who feature in Europe's top-five leagues, the exception being Sarr of Watford.
But the story could have been very different had the referee pointed to the spot for Sarr's challenge on Afif, but he left the locals outraged by failing to award a penalty.
It proved crucial as Khoukhi's scuffed clearance allowed Dia to convert the opener, and Diedhiou's unstoppable header double the advantage on 48 minutes.
Qatar, though, showed good fight to hit back via Muntari's thunderous header, which sent the partisan crowd into raptures.
But any optimism for an equaliser was killed when Dieng finished off a swift move to seal Senegal's first victory in the group.
The Qatar players were applauded by their fans at the final whistle, but some hit the deck in regret knowing their prospects of reaching the knockout stage are now over.
-- Courtesy of BBC Sport