Howe's men went behind to Scott Dann's goal on 27 minutes, but equalised seven minutes later through Marc Pugh. Club record signing Benik Afobe secured the win shortly before the hour to send Bournemouth up one place to 15th.
"When we went a goal down we had to show real resilience. We weren't very good in the first half but that goal from Marc Pugh was important, to score so quickly after," Howe told reporters.
"It gave us a life line because we were second best in the first half and dipped below our usual levels. But we responded brilliantly and, I felt, just about deserved it in the end."
Bournemouth have a seven-point cushion over the teams in the relegation zone and Pugh said the south-coast side can keep on improving in their first season in the top flight.
"We really needed that result. It gets us up the table and we can push on again," he said.
"We're on a great run, we've beaten Chelsea and Manchester United, world class sides. We're going from strength to strength, we showed plenty of belief and got our reward."
Bournemouth host fourth-placed Arsenal on Sunday.