Looking to avoid a fourth consecutive home defeat, the Blaugrana calmed a nervous Camp Nou crowd through Depay's excellent first-half volley on Monday (AEST).
Busquets then fired a low effort beyond Sergio Rico and the Catalan giants held on for the win despite Antonio Raillo's 79th minute goal.
After being forced to watch rivals Real Madrid win the title on Saturday, Xavi's side at least look likely to join them in next season's Champions League after moving back into second place.
Rico was forced into an excellent save after five minutes, diving to his right to turn away Pierre-Emerick Aubameyang's goal-bound header, before Fer Nino squandered a fantastic opportunity for the visitors when firing over at the back post from close range.
Ronald Araujo saw a powerful 22nd-minute header ruled out for offside, but the Blaugrana did take the lead three minutes later when Depay expertly volleyed into the bottom-left corner after controlling Jordi Alba's ball over the top.
The host lost Gerard Pique to injury before the break, but doubled their advantage after 54 minutes when Busquets drilled a left-footed effort into the bottom-right corner after striding onto a loose ball on the edge of the area.
Depay went close to another brilliant goal when he shot narrowly wide from range after dancing through the visiting midfield, before Barca had another goal ruled out by VAR after Ferran Torres's shot deflected in off Aubameyang in an offside position.
Xavi handed Ansu Fati his first appearance since November from the bench, as the hosts clung on for a vital win despite Raillo prodding home from Salva Sevilla's cross.