The Cobblers had to be content with a draw after having the better of a dull match without many clear openings.
For Bournemouth, the failure to win now means no victories in six matches since Sean O'Driscoll left to join Doncaster.
The Cherries struggled from the start with experienced midfield player Marcus Browning booked for a foul after just nine minutes of his first home game since returning from suspension.
Northampton should have done better with their opportunities, although they were missing seven experienced players and on the bench had three youngsters without previous league action.
After an even first half it was Northampton who were on top in the second period.
McGleish should have been the match-winner and his worst moment came in the 53rd minute.
Mitchell Cole, on loan from Southend, crossed from the left to McGleish six yards out, but he failed to control the ball and saw it rebound off him to safety.
Eoin Jess went close with a free-kick, tipped over by Gareth Stewart, and Jason Crowe cut in from the left to unleash a right-foot shot which flashed narrowly wide.
McGleish went close again in the 74th minute with a header from a centre by Crowe which came back off the post.
Bournemouth were also saved by goalkeeper Stewart, who twice made brilliant saves to deny Andy Kirk when clear.
The hosts' best chance came in the 78th minute when Stephen Purches saw a fierce free-kick cleared off the line by Crowe.
There was little chance for Darren Anderton to show his England class and he limped off after 52 minutes.
Anderton's brightest moment was a stunning centre from the right in the first half which leading scorer James Hayter failed to connect with ten yards out.
Bournemouth's acting manager Stuart Murdoch sent on Steve Fletcher for the last ten minutes, but the big striker could make little impression and Northampton held on for a draw.