Kai Havertz headed Arsenal into the lead in the 37th minute, taking the title-thirsty club one step closer to its first Premier League title in 22 years.
If Manchester City drops points away to Bournemouth on Tuesday, Arsenal will be crowned champions.
Should City win, however, the gap would be two points going into Sunday's final league round, when Arsenal can clinch the title themselves away at Crystal Palace while City host Aston Villa.
Arsenal and national-team striker Viktor Gyökeres started on the bench against Burnley, but came on in the 73rd minute as Havertz was substituted.
