Begovic made two excellent first-half saves as Rangers were unlucky to leave the south coast with only a solitary point.
Bournemouth staved off interest from West Ham and Torino to keep striker Josh King during the transfer window but the Norwegian was once again conspicuous by his absence.
The Cherries were strangely subdued for much of the afternoon and indebted to former Chelsea star Begovic for keeping the scoresheet blank.
It took just four minutes for Bright Osayi-Samuel to call Begovic into action.
Osayi-Samuel cut in from the right wing before shooting left-footed and forcing a sprawling save from Begovic at his near post.
Begovic showed quick reactions moments later to come racing off his line and nick the ball from the toes of Macauley Bonne after the former Charlton striker had got in behind Bournemouth's defence.
Bonne looked certain to open the scoring five minutes later only for Begovic to thwart him again.
It took until 11 minutes before half-time for sluggish Bournemouth to create a meaningful chance.
Chris Mepham launched a hopeful punt upfield which allowed Dominic Solanke to spin in behind the Rangers defence and his half-volley flashed wide of the far post.
Rangers continued to look the more likely to break the deadlock in the early stages of the second half and the lively Osayi-Samuel drilled a right-footed effort into the side-netting.
Dan Gosling netted twice in the Cherries' 3-1 win at Coventry before the international break and the former Everton midfielder's powerful 64th-minute shot was blocked bravely by Rangers captain Geoff Cameron.
Solanke should have done better 11 minutes from time after latching onto Lewis Cook's lofted free-kick but buried a tame shot into the side-netting.
Bonne curled a 20-yard shot over Begovic's crossbar with eight minutes left to play as Rangers pressed in vain for a late winner they would have deserved.