Cherries top scorer Solanke gave his side the lead just before half-time with his 21st league goal of the season, and substitute Lowe secured the win in the 90th minute.
Story of the match
Derby youngster Malcolm Ebiowei had the first chance of the afternoon. The 18-year-old skilfully beat two Bournemouth players but saw his effort comfortably saved by Mark Travers.
Soon after, the Rams had a penalty appeal turned down by Andy Madley when Ravel Morrison appeared to have been fouled by Leif Davis.
Ebiowei's cross in the same attack found Luke Plange, who failed to hit the target.
Bournemouth were forced into a change just before the half-hour mark, with Jordan Zemura replacing the injured Davis.
Allsop was called into action again in the 36th minute. Anthony's cross was headed back across goal by Ryan Christie, and the Rams keeper made a crucial punch to beat Jefferson Lerma to the ball.
Derby had been defending well but had to defend a throw-in with ten men after Lee Buchanan received treatment for the second time in the game.
Wayne Rooney's side were helpless on this occasion.
Half-time: Bournemouth 1-0 Derby
Solanke almost doubled his side's advantage soon after half-time, but his header from Christie's corner was expertly saved by Allsop.
Buchanan then forced a save from Travers at the other end after being found by Rams skipper Tom Lawrence.
The ball dropped to Davies from the resulting corner, but Zemura made a superb block to deny him.
Morrison was denied by Travers in the 62nd minute, with Lawrence again the architect.
Christie should have made it 2-0 to the hosts 15 minutes later, but his strike from inside the box was straight at Allsop.
But the Cherries did get a second late on.
Solanke's shot was saved by Allsop, who was unable to stop Lowe from scoring the rebound.
AFC Bournemouth: Travers; Stacey, Phillips, Kelly, Davis (Zemura 29); Cantwell (Lowe 66), Cook, Lerma; Christie (Pearson 90+3), Solanke, Anthony
Derby County: Allsop; Byrne, Davies, Cashin, Buchanan; Bielik (Ebosele 67), Bird; Ebiowei (Knight 76), Morrison, Lawrence; Plange (Kazim-Richards 66)