Kiwi pair help send La Rochelle top
Scrumhalf Kerr-Barlow scored the opening try after Gaetan Germain had given the relegation-threatened visitors a third-minute lead.
Fijian Benito Masilevu restored Brive's lead with a try but French centre Elliot Roudil hit back for the hosts.
Club captain Eaton dotted down a cross kick in the corner on the stroke of halftime and Roudil bagged his second after the break.
But two tries from former Toulon wing Benjamin Lapeyre denied La Rochelle an attacking bonus.
Montpellier can return to the Top 14 summit if they win on Sunday when New Zealander coach Vern Cotter returns to former club Clermont for the first time.
Fijian powerhouse wing Josua Tuisova was sent off but Toulon still entrenched themselves deeper into a play-off spot with a 36-12 victory over Bordeaux-Begles.
Rugby League convert Semi Radradra, another Fijian, bagged two tries and former England international Chris Ashton also crossed the whitewash as Toulon's six-try performance earnt them a bonus point that moved them up to fourth.
"We have to be satisfied with this match. We won it with 15 against 15 and we drew with 14 against 15," said Toulon head coach Fabien Galthie.
"We haven't given up in any match since the start of the season and this win is important for us."
Tuisova also scored a try just three minutes before he was dismissed for a dangerous high tackle on Bordeaux's Australian wing Blair Connor.
Forwards Juandre Kruger and Raphael Lakafia rounded out the Toulon tries although Francois Trinh-Duc managed to convert only three from six.
Connor and Romain Lonca scored consolation tries for the visitors, who had started the day only a point behind their hosts.
The game was over as a contest before Tuisova was dismissed as Radradra sliced through a porous Bordeaux defence and Ashton finished off a sweeping counter for his 14th try of the season, both inside the first 10 minutes.
Bordeaux lost centre Jean-Baptiste Dubie and full-back Geoffrey Cros to injuries inside the first 20 minutes, the latter's knee injury looking particularly shocking.
"It seems quite serious, it looks like a (torn) cruciate knee ligament," said Bordeaux president Laurent Marti.
Toulon were without mercy as Lakafia touched down on the base of the post before Tuisova found space to notch a fourth, giving Toulon a 26-0 lead.
But the Fijian was then sent off, giving Bordeaux hope, although the hosts held their own with the numerical disadvantage, sharing four more tries with their visitors.
Elsewhere Toulouse and Lyon kept in the play-off picture with respective victories over Oyonnax and Agen – the bottom two.
Toulouse were made to graft for long periods by rock bottom Oyonnax but late tries from Louis-Benoit Madaule and Leonardo Ghiraldini helped them snatch a bonus point in a 37-15 success.
However, France veteran lock Yoann Maestri, was sent off for fighting, along with Oyonnax's New Zealander No.8 Rory Brice – the pair even continuing their scrap as they left the field.
Australian international flank Liam Gill got the ball rolling for Lyon with a second minute try in a 10-try 71-17 hammering of sorry Agen.
Lyon were the leaders eight games into the season before a run of five straight defeats derailed their play-off charge.
But this bonus point romp, including a hat-trick from scrumhalf Baptiste Couilloud and 20 points from flyhalf Lionel Beauxis keeps them on the cusp of a top six position.
Saturday's results:
La Rochelle 33 Brive 24
Lyon 71 Agen 17
Toulon 36 Bordeaux-Begles 12
Toulouse 37 Oyonnax 15
Castres 13 Racing 92 18
Sunday's fixtures
Stade Francais v Pau
Clermont v Montpellier
Agence France-Presse
