Albion took the lead just after half-time through Jay Rodriguez but a bad mistake by Ben Foster allowed Jordan Ibe to bring the home side level and a superb curling free-kick by Junior Stanislas a minute from time sent the visitors to their eighth straight defeat and nearly certain relegation.
Alan Pardew was handed the reigns at the Hawthorns in November, but his side have managed to win just once since then, and relegation to the Championship is starting to look like a very real possibility.
Rodriguez slammed home in the 49th minute to put West Brom on track for a significant three points that would have sparked life into their bid for survival.
Bournemouth climbed one place, to 11th on 36 points, eight clear of the relegation zone.
Begovic was the busier of the two keepers, though, and he denied a long-range effort from Rodriguez before also getting down well to deny another glancing header from Rondon shortly before the half-hour mark.
Francis' replacement Charlie Daniels had a 22nd-minute effort easily saved by Foster as both sides struggled to create clear chances.
JUNIOR STANISLAS struck a stunning free-kick victor as rock-bottom West Brom let a lead skip again.
West Brom scored the goal their first-half performance deserved shortly after the restart.
But Tosun had the final word with an agile diving header in the 84th minute.
It was Ibe who came up with the goal for the hosts, though, the winger sending an ambitious shot at goal that Foster should have dealt with, and the goalkeeper's mistake gave Bournemouth hope they could win the game.
"We got ourselves in a great position, made a couple of errors and got punished for it".
West Brom are in desperate need of a win ahead of Saturday's league trip to Bournemouth as the Baggies lie eight points adrift of safety with as many matches left to play. Three of the others are against teams in the relegation scrap with them, Crystal Palace, Swansea City, and Southampton - who just fired their manager Mauricio Pellegrino.