Gouiri hat-trick guides Marseille back to second in Ligue 1
Mason Greenwood was also on target with his 19th league goal of the campaign as Marseille made it back-to-back home wins with nine goals scored in eight days.
Roberto De Zerbi's side are second with three games left, 20 points adrift of champions Paris Saint-Germain and two points ahead of third-placed Lille, who won 2-0 at struggling Angers earlier.
However, just four points separate Marseille from Strasbourg in seventh, with six sides locked in an intense fight for European places as the French season reaches its climax.
The top three teams in Ligue 1 will qualify directly for the league phase of next season's Champions League, with fourth entering in the preliminary rounds.
Fifth will go into the Europa League and sixth will also if PSG beat Reims in next month's French Cup final, while a final spot is on offer for the Conference League.
Marseille are desperate to qualify for Europe's elite club competition and spent most of the last week on a training camp in Italy, based just outside Rome.
The decision to take the squad away paid dividends, even if Algerian international Gouiri's early opener was cancelled out before the half-hour mark as Senegal's Abdallah Sima finished from close range for Brest.
Greenwood restored Marseille's lead with a fine right-foot strike from just outside the area, before Gouiri took centre stage as he made it 3-1 on the stroke of half-time.
When Greenwood held up a cross from the byline, Gouiri controlled on his chest before readjusting his body to score with a wonderful acrobatic overhead kick.
He then ran through to complete his hat-trick and make it 4-1 in the second half with his seventh goal in 11 games since joining OM from Rennes in January.
Lille went third as goals either side of half-time from Alexsandro and Hakon Haraldsson gave them a third consecutive win and all three points at Angers, who are just above the relegation zone.
Lille host Marseille next weekend in a huge game, the outcome of which could benefit the teams clustered just behind them in the table.
Monaco in fourth were held to a 1-1 draw at relegation-threatened Le Havre on Saturday, while Lyon, Nice and Strasbourg all won over the course of the weekend to stay right in contention.
Elsewhere on Sunday, Auxerre won 4-0 at Lens to end the European hopes of Will Still's team.
Nantes drew 0-0 at Toulouse to edge closer to safety, while Reims did likewise with a goalless stalemate at Montpellier, whose relegation was confirmed on Saturday.