Victories for fellow strugglers Swansea City and Hull City elsewhere on Saturday afternoon means that Steve Agnew's men are now nine points adrift of safety, with just one game in hand to play on the two sides directly above them.
A disastrous day for Boro, which surely all but ends their one-year stay in the top flight, began with an in-form King tapping in from Pugh's cross less than two minutes into the match.
Chances went begging at either end for King and Alvaro Negredo, but any outside hope the Smoggies - the only team in European football's top five leagues without a win in 2017 - had of finding a route back into the match were soon over.
A second yellow card for Ramirez, this time for bringing down Pugh, brought an early end to his afternoon and gave the 10 men of Boro an almighty mountain to climb.
Afobe and substitute Lewis Cook both tested Guzan in quick succession soon after the restart, while Ryan Fraser was also kept out by the American in spectacular style, but Boro could not hold on for much longer as Pugh curled one into the far corner on his 250th league outing for the Cherries.
The fourth was added by Daniels, who skipped past a couple of players and rifled home from 20 yards as Bournemouth maintained their unbeaten home league record against their opponents in some style.
There was a late chance for Lys Mousset to put the icing on the cake after being brought off the bench, only to blaze over the bar from six yards out with the final opening of the one-sided affair.