Andoni Iraola's Bournemouth picked up their first away point of the season but will feel hard done by after a positive display at Gtech Community Stadium.
And the hosts converted their opportunity and took a 1-0 lead in the seventh minute. Jensen looked set to cross the ball into the box but his quick thinking saw the Dane shoot at Neto's near post, the Brazilian scrambled in an attempt to keep out the effort but VAR ruled the ball had crossed the line.
The Cherries sought a leveller through Ryan Christie in the 18th minute. The Scot slalomed his way through red and white shirts but his run was bizarrely intercepted by team-mate Solanke before he could test Mark Flekken from yards out.
And, testament to Frank's fast-paced football, Brentford instantly responded through Vitaly Janelt who picked out Schade, who was only prevented from doubling the lead by a post.
The end-to-end nature of the game continued in the 30th minute as the Cherries levelled through Solanke.
Marcos Senesi returned to Iraola's starting XI and the Argentinian's curved through-ball found the composed Solanke, who beat his marker before finishing in front of the away fans.
Frank's decision to proceed with a back-four for a third consecutive league game paid dividends as midfielders Jensen, Janelt and Christian Norgaard held the lion's share of first-half possession in a frantic match.
The hosts were quick out of the traps in the second half when Rico Henry gambled on the loose ball and expertly picked out the dangerous Mbeumo, who fluffed a golden chance to retake the lead when he shanked his effort wide.
Henry started a similar move in the 59th minute but Yoane Wissa's missed shot was a reminder of how much Brentford needed the suspended Ivan Toney, on a day when they were starting to rue missed chances as Bournemouth gained momentum.
Flying full-back Henry's 74th-minute corner found the head of Norgaard, who forced a stretched save from Neto.
Henry's loose backpass was cruelly intercepted by the visitors three minutes later and, after some neat play in the area, Brooks scored for a second successive game to give Bournemouth a 2-1 advantage.
The home crowd demanded a response and they got it in added time through winger Mbeumo, who was the first to react to Nathan Collins' long ball, earning the hosts a point.