The Saints kept Bournemouth at bay for just 10 minutes, as Ake beat Kevin Danso in the air from a Diego Rico corner to power a header home.
Joshua King thought he had made it 2-0 just before the half-hour mark, only for a VAR (video assistant referee) review to disallow it for offside.
Bournemouth did not have to wait much longer, however, as Wilson guided Philip Billing's cut-back into the bottom-right corner with 35 minutes played.
Saints pulled one back from the spot through James Ward-Prowse early in the second half after Steve Cook had clumsily tripped Che Adams.
Bournemouth was controversially denied a spot-kick of its own when Cedric Soares sent King tumbling after an hour.
Although the Saints piled the pressure on towards the end, a calamitous mix-up between goalkeeper Angus Gunn and Jan Bednarek in stoppage-time allowed Wilson to wrap up a first ever win away to Southampton.