Leighton Baines secured the points with a well-struck second-half winner but the full-time whistle prompted many banners to be unfurled calling for Martinez to be sacked.
During the game an aeroplane circled above Goodison Park trailing the message: "Time to Go Roberto".
Everton's slide was at least halted by three overdue points, but a week after an FA Cup semi-final defeat and with a hammering by Liverpool fresh in the memory, Martinez's future still looks to be in serious jeopardy.
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"The win was very important -- it has been an emotional time. We had to win. We had to dig deep, we showed we were a team and that we have character," the Spaniard said.
Asked about the protests, he added: "Those aspects are not for me. My concentration is preparing the players and preparing for the game."
A clash between teams with little to play for apart from pride began in lively fashion with two goals in the opening 10 minutes.
First Tom Cleverley fired Everton ahead with a neat turn and shot but Bournemouth responded almost immediately with Marc Pugh rolling the ball home from close range.
Fears of a ninth home defeat in all competitions this season for Everton subsided, however, when Baines scored the winner, slamming home Aaron Lennon's cross.
(Reporting by Martyn Herman, editing by Pritha Sarkar)
