Morrison's goal added to those of Chris Brunt and Gareth McAuley saw the hosts come from 1-0 down at half-time -- Robert Snodgrass's seventh of the season -- and looking distinctly second best to extend Hull's away win drought to nine.
West Brom remain in eighth place in the table whilst Hull -- who have won just once in their last 18 Premier League games -- are second from bottom four points adrift of safety.
Their inability to keep a clean sheet has also hurt them with this match making it 22 in all competitions without a clean sheet -- their worst run since January 1991 when they also conceded in 22 successive games.
Hull had looked in good shape when Snodgrass put away a neat half volley from Ahmed Elmohamady's cross for their first away goal in six matches.
It was no more than the visitors deserved and emboldened by their 2-2 draw with Everton on Friday they had started the brightest with Snodgrass, Adama Diomande and skipper Michael Dawson going close.
West Brom sprang to life when Cameroon defender Nyom came on for the injured Jonny Evans -- his cross finding leading scorer Salomon Rondon but the Venezuelan headed tamely straight at David Marshall.
Marshall -- whose howler of an own goal on Friday gifted Everton their first equaliser -- made a cracking save a few minutes later getting down superbly to stop in-form Matt Phillips' diving header.
Hull went down the other end straight away with Diomande in a terrific spot to shoot but McAuley did a brilliant job in blocking his strike, the ball flew out to Hull midfielder Tom Huddlestone whose shot was tipped away by Ben Foster for a corner.
West Brom manager Tony Pulis must have given his players a roasting at the break because they came out a side transformed and went level within four minutes.
Phillips's corner was met by an unmarked Brunt at the back post and he made no mistake with his header for his second of the season.
The hosts barely allowed Hull a touch and it was little surprise when they went ahead in the 62nd minute McAuley's effort from a corner adjudged by Mark Clattenburg to have crossed the line despite being stopped by Sam Clucas.
West Brom should probably have been punished a few moments later as Elmohamady found Ryan Mason in acres of space in the penalty area but the former Spurs man sent his header harmlessly over the bar.
His miss proved even costlier as the hosts ended the match as a contest almost straightaway Morrison slotting home from 10 yards out after being teed up by Rondon.