The home side AFC Bournemouth, recorded 13 total shots, while the visiting Blues managed one additional shot, with a total of six on target.
In the third minute, Chelsea took a shot on goal when the day’s captain, Gallagher, saw his attempt deflected for a corner kick.
Mauricio Pochettino’s team maintained a high tempo throughout the first half, while Bournemouth remained on the defensive.
In the 50th minute, Raheem Sterling saw his goal disallowed for an offside after his well-executed free kick hit the crossbar.
Bournemouth had a promising opportunity in the 78th minute when Billing unleashed a powerful strike that deflected off Gallagher.
Neto and Sanchez, the goalkeepers for their respective teams, were consistently summoned to make crucial saves throughout the match.
Both teams shared the points in a goalless draw, with Bournemouth remaining in 15th place.
Chelsea sits in 14th place on the league table, enduring a winless streak of five matches, following their home defeat to Nottingham Forest in their most recent EPL fixture.