Romelu Lukaku scored twice as Belgium recorded their first win at Euro 2016 and damaged the Republic of Ireland's chances of reaching the last 16.
After a goalless opening half, Lukaku scored the Belgians' first goal at the tournament with a neat finish from the edge of the area.
Axel Witsel headed in a second in the 61st minute, before Lukaku sidefooted home after a quick counter.
The Republic did not force goalkeeper Thibaut Courtois into a single save.
It means Martin O'Neill's side, who are bottom of their section with one point, will probably have to beat Italy in their final Group E game on Wednesday.
Belgium, meanwhile, are second on three points and next face Sweden.
It was a disappointing performance from the Republic after a promising display in their opening fixture, when they drew 1-1 with Sweden.
Shane Long, operating as a sole striker, found himself isolated for long periods but may feel he should have won a penalty when he was caught by a wild Toby Alderweireld boot inside the Belgium area, just before Lukaku opened the scoring.
Belgium boast a wealth of individual talent and had topped the Fifa rankings within the last year, making them one of the favourites to go far in France.
However, they looked nothing like a team capable of winning the tournament against Italy, against whom they lost 2-0 on Monday.
The Belgians lacked fluidity and a sense of unity. As goalkeeper Thibaut Courtois put it: Tactically, technically and organisationally, they came up short.
It was a different story against the Republic.
They dominated possession throughout, with Eden Hazard and Kevin de Bruyne proving much more effective at stretching their opponent's defence than they were against the Italians. Both finished with an assist each.
The win means they are in a strong position to finish second in the group and, while they will face opponents much more attack-minded than the Republic, the victory will undoubtedly boost a side that had appeared lacking in confidence at the start of the tournament.