The under-fire manager will be left to rue Wilfried Bony's disallowed goal on the stroke of half-time, but neither side did enough to win a game desperately lacking in quality.
Ivory Coast international Bony whipped a wonderful effort in off the underside of the crossbar, but his celebrations were cut short by referee Stuart Attwell, who deemed Jordan Ayew's challenge on Nathan Ake in the build-up to be too aggressive.
Leroy Fer squandered a glorious opportunity to win it with seven minutes remaining when he inexplicably hooked wide after his initial shot had been saved.
Swansea's recent woes were not evident in the early stages and they came close to taking the lead inside three minutes as Ayew's cross from an acute angle crashed off the outside of the post.
Bournemouth lacked attacking intent in the opening half hour and had only a long-range effort from Andrew Surman, which was comfortably held by Lukasz Fabianski, to show for their endeavours.
Chances were at a premium throughout the opening period, but the hosts did carve out two in the space of 60 seconds 10 minutes before the break.
First, Martin Olsson skewed a volley over the crossbar after Tom Carroll had clipped a clever ball over the Cherries' defence and then Renato Sanches drilled into the side netting from just inside the area.
Joshua King fired straight at Fabianksi after being found in the area by Marc Pugh moments before Bony's superb strike was controversially ruled out.
Adam Smith fired wide shortly after the break as Bournemouth suggested they would adopt a more expansive approach in the second period.
That hope was short-lived, though, as the game settled back into its first-half pattern of stodgy midfield scrapping.
The players' frustrations boiled over with 15 minutes remaining as Ki Sung-yueng and Jordon Ibe exchanged pushes, with both receiving yellow cards.
Substitute Fer then had the chance to claim a priceless three points late on. The Netherlands international burst through the Bournemouth backline and after his first attempt was kept out by Begovic he clipped the rebound wide with the Bosnian goalkeeper stranded.
Key Opta facts:
- Swansea City stopped a run of four consecutive losses in the league, but they are without a win in their last five (D1 L4), their longest winless run since April 2017 (six).
- Bournemouth have kept three successive Premier League clean sheets for the first time.
- The Swans have failed to score in three successive Premier League games for the first time since May 2013.
- The Swans recorded just one shot on target and have managed just 24 shots on target in the Premier League this season; 10 fewer than any other side.
- Wilfried Bony has now gone nine Premier League games in a row without a goal. The Ivorian striker has only netted in one of his last 27 Premier League games (two goals).