Morris came off the bench to score for the fourth time in the last month as his 80th-minute strike settled a topsy-turvy Championship game in the Tykes' favour.
Bournemouth had recovered from Michal Helik's early opener to lead at the break courtesy of goals from Arnaut Danjuma and Dominic Solanke, but Dominik Frieser levelled before Morris struck the winner.
The win was Barnsley's eighth in nine games, extended their unbeaten league run to 11 games and opened up a five-point lead over their beaten hosts.
Barnsley took the lead against the run of play in the 16th minute.
Bournemouth hit back just six minutes later when Danjuma swept in his ninth goal of the season from close range.
Goalkeeper Asmir Begovic started the move, throwing the ball half the length of the field to Junior Stanislas who knocked it through the legs of Jordan Williams before cutting his cross back for Danjuma to slot home.
Stanislas thought he had put Bournemouth in front in the 28th minute but his thunderous right-foot volley was somehow tipped on to the crossbar by Collins.
Jefferson Lerma's goalbound effort from just inside the penalty area was then diverted behind for a corner by Helik.
Bournemouth carved another good chance in the final minute of the first half when Stanislas floated in a teasing free-kick but Solanke could only manage to head tamely off target.
Solanke made amends seconds later by sliding in at the far post to turn in Danjuma's perfectly-weighted pass for his 11th goal of the season.
Kane almost made an instant impact by driving a long-range shot into the gloves of Begovic.
The visitors had a huge let off in the 58th minute when Collins came racing out of his penalty area and miscued a clearance but Solanke could not quite squeeze the ball into an empty net from a tight angle.
Barnsley were back on level terms two minutes later when Frieser finished neatly from 10 yards after a Cauley Woodrow cross had been deflected into his path.
Tykes substitute Morris then headed wide from captain Alex Mowatt's pin-point cross when he should have at least hit the target.
Kane also drew a good flying save from Begovic with a curling free-kick 17 minutes from time.
Valerien Ismael's side took a deserved lead in the 80th minute when Morris grabbed his fifth goal since being recalled from a loan spell at MK Dons.
Morris towered above a static Bournemouth defence to meet Mowatt's free-kick and send a looping header into the top left corner of Begovic's goal.