Bolton striker Kevin Davies had two late efforts ruled out for offside, while Baggies' Kim Do-Heon saw his 78th-minute thunderbolt rattle the underside of the crossbar before goalkeeper Jussi Jaaskelainen brilliantly save Ishmael Miller's follow-up header.
Tony Mowbray's newly-promoted side were searching for their first win of the season, and they were guilty of wasting several gilt-edged chances to take all three points.
Both sides came into the game on the back of humiliating midweek Carling Cup defeats at the hands of Northampton and Hartlepool respectively.
But it was West Brom that went closest to taking the lead in an uninspiring first half when Jaaskelainen produced a stunning one-handed save to tip behind James Morrison's curling effort from 20 yards.
The Bolton keeper was back in action to deny Borja Valero after the Spaniard, a £4.7million signing from Real Mallorca, had burst into the area unchecked.
Gianni Zuiverloon should have given the visitors the lead five minutes after the restart but the Dutch Under-21 international headed Jonathon Greening's free-kick over from six yards.
The Baggies wasted another glorious opportunity to take the lead when Miller ran onto Paul Robinson's long ball over the top but the former Manchester City striker blazed over.
Bolton, and their fans, finally came to life when Scott Carson made a brilliant save to keep out Riga's free-kick which was flying in to the top corner on 63 minutes - and Kevin Nolan headed Jlloyd Samuel's cross wide minutes later.
But Gary Megson's side had a huge let-off when the impressive Morrison dragged his shot across the face of goal.
Kim was desperately unlucky not to give West Brom the lead with 12 minutes left, although they received a major let-off a minute later when Greening looked to have handled inside the area.
Davies then had the ball in the back of the net twice in the last four minutes - only for both efforts to be disallowed by the assistant referee's flag.