Andoni Iraola's side claimed their first-ever win at Old Trafford, after thumping Manchester United 3-0 away from home, to extend their unbeaten run to five matches in the Premier League.
The visitors were rewarded for their fast start, with Lewis Cook winning the ball high in the opposition half before playing in a low cross to Solanke, who scored his eighth goal of the Premier League season, placing the ball past Andre Onana in net.
Bournemouth looked comfortable heading into the break 1-0 up and could have added another, with Marcus Tavernier having a goal disallowed for offside.