Boro held a 67-year unbeaten run against the visitors and were the better side in the opening half an hour, having unsettled Wolves with some physical tackles and high pressing.
Despite largely being on the back foot, Wolves themselves boasted a couple of long-range Neves shots which worked Darren Randolph in the same period, and then turned the momentum of the match on its head as Helder Costa and Ivan Cavaleiro hit Boro with a quick double sucker-punch.
Costa thumped a Barry Douglas cross home at the far post on 32 minutes, after Wolves had quickly recycled the ball following Randolph's initial save from the goalscorer.
Randolph was at full stretch again to push away Willy Boly's glancing header five minutes later, only for the opportunistic Cavaleiro to nod home the rebound from a couple of yards out.
There was not much goalmouth action in the second half but there was plenty for referee Stuart Attwell to contend with, starting with a blatant trip by last man George Friend on Costa which went unpunished.
This incident riled Wolves and Neves allowed his temper to flare with late tackles on Adama Traore and Friend in the space of a couple of minutes before the hour mark, which consigned him to an early bath.
Doherty followed Neves down the tunnel for a pair of fouls on the same Boro duo, the second of which was an apparent elbow on Friend which left Wolves two men down with nearly a quarter of the game to play.
The Boro onslaught took a while in coming and when it arrived, Jonny Howson and Stewart Downing both lacked composure from the edge of the box, and Ruddy saved well from Traore diving low at his near post.
Bamford's stunning finish with the outside of his left foot on 94 minutes came too late, although Downing could have snatched Boro a point when he dragged wide from 20 yards out with Ruddy in no man's land.
Boro remain in the final playoff place but only two points separates them from Preston North End in 10th. Meanwhile, three more wins for Wolves will seal their return to the Premier League after a six-year absence.
MIDDLESBROUGH (4-5-1): Randolph; Shotton (Cranie, 70'), Ayala, Gibson, Friend; Clayton (Assombalonga, 70'), Leabitter (Howson, 63'), Besic, Traore, Downing; Bamford
WOLVES (3-4-3): Ruddy; Boly, Coady, Bennett; Douglas, Saiss, Neves, Doherty; Cavaleiro (N'Diaye, 58'), Costa (Gibbs-White, 84'), Afobe (Bonatini, 63')