A Marko Arnautović penalty gave the hosts the early advantage after Nathan Ake was adjudged to have fouled Javier Hernández in the penalty area, but star striker Callum Wilson levelled the scoreline by sending the ball through the goalkeeper's legs after a mazy run.
The reliable Steve Cook then popped up with what proved to be the winning goal in the second half by converting a simple free kick routine, with the Cherries seeing out the rest of the match to ensure they returned to the south coast with six points from their opening two games.
"It was a really good win. When we conceded it was at a spell when West Ham were on top and we found it difficult, but credit to the players second half," Howe said, quoted by Sky Sports.
"We had a number of chances to score. It's a great comeback. Whenever you come here and win it's a real achievement.
"There were loads of chances that we didn't take and you think, 'Is it going to be one of those days?', but thankfully Callum's goal was the turning point. It was a great goal from him individually. That was the boost we needed to go on and win the game."
Bournemouth registered a total of 21 points last season from losing positions and the Cherries once again showed their determination by clawing three points from the jaws of defeat against West Ham.
Howe's side will now return to Dorset in preparation to welcome Marco Silva's impressive Everton to Dean Court, with the Toffees looking to maintain their unbeaten record next week.