Bournemouth wasted a number of chances as they recorded their second goalless draw in a row.
Bournemouth eased into their neat passing game early on and striker Dani Rodrigues knocked down for Stephen Cooke who volleyed just wide of the post on eight minutes.
Rodrigues then blew a golden chance on 20 minutes when he volleyed wide after good work by Andrew Surman.
Tranmere defender Ian Goodison got in a muddle moments later and Surman tried his luck from distance only to see the ball fizz past the post.
After the break Achterberg stood before Bournemouth and the opening goal when he produced a wonder save on 51 minutes.
Rodrigues got on the end of Brian Stock's precise cross only to see his header brilliantly pushed away by the keeper.
And man of the match Achterberg was again Tranmere's saviour on 64 minutes.
Influential winger Surman swung in a low cross which fell to Marcus Browning whose powerful volley was somehow beaten away.
Tranmere posed little threat going forward and took 83 minutes to register a shot on target.
Midfielder Sam Aiston cut in from the left before unleashing a right-foot curler which Bournemouth goalkeeper Gareth Stewart had to be alert to put over the bar.
Two minutes from time the visitors did have the ball in the net through Chris Greenacre but his effort was chalked off for offside as the game ended goalless.