Cook scored from close in at the second attempt after seeing his initial effort blocked following a corner.
The Cherries had chances before the goal with Andrew Surman missing the target from a good position and the woodwork coming to the home's rescue late on to deny Matt Ritchie. Barnsley goalkeeper Luke Steele also made some good saves.
Howe said: "I felt we deserved to win, overall.
"Barnsley started the game well and had a couple of chances themselves in the game, but overall I thought we were the dominant team and had more chances.
"Barnsley are fighting for their lives and I feel for them because it was a cruel one but I'm really pleased for my team and my supporters.
"We talk to our defenders all the time about set-pieces and Ian Harte's delivery again was so important.
"Our defending in recent games has been a highlight and we defended very well as a group.
"It pleases me no end to get clean sheets because it's not something we've been associated with this year, but that has been a real improvement in the side in recent weeks.
"I'm reluctant to talk about individuals but Steve Cook put in a good performance and Tommy Elphick led by example as captain. The whole back four has played a huge part in that.
"If the season finished now, I'd say that we'd achieved our objective.
"We're in a good position to move forward with a young team to get better next year."
- Sports & Recreation
- Andrew Surman
- Marcus Pedersen
- Martin Woods