The Baggies had by far the better of the chances in both halves, with Peter Odemwingie squandering three great opportunities and Chris Brunt heading against a post.
But they simply could not find the net and, after Taylor gave Wanderers the lead with his first goal of the season in the 40th minute, Elmander made the points safe four minutes from time.
Defences were certainly on top in the early stages and it took until the 16th minute for the first shot of the match, Graham Dorrans firing straight at Jussi Jaaskelainen.
Former Baggie Paul Robinson then had to be alert to head away a Somen Tchoyi cross before, as the game opened up, a flowing Bolton move ended with Elmander shooting over the bar when at the very least he should have made Scott Carson work.
West Brom were looking dangerous, though, and, after Tchoyi had sent another effort wide, leading scorer Odemwingie beat the offside trap to latch onto James Morrison's deft chip only to lose his bearings and shoot well wide from eight yards.
The Baggies were certainly on top, though, and they should have gone in front 10 minutes before half-time when Odemwingie again beat the Bolton offside trap.
A simple square ball to Graham Dorrans would have surely seen the visitors go one up but his pass was weak in the extreme, earning the striker an earful from Brunt.
And that missed opportunity was made to look even more costly when Taylor gave the hosts a lead they scarcely deserved in the 40th minute.
Taylor was both architect and scorer, freeing Lee with a neat chip. The Korean looked like he had taken it too far after going round Carson but he had the presence of mind to look up and play in Taylor, who tucked away his first of the season.
West Brom sought an immediate leveller and Gonzalo Jara forced Jaaskelainen to tip a shot over his bar before Gary Cahill was lucky to get away with one touch too many in the area.
West Brom continued their goal quest after the break with a succession of chances that began when Tchoyi struck a fierce shot at Jaaskelainen from a tight angle.
The visitors came even closer moments later, and again Tchoyi was at the heart of the move. The midfielder dug out a cross that Brunt got to ahead of Jaaskelainen only for his header to cannon back off the post before being hacked away.
Brunt then fired an effort into the side-netting before the midfielder and Odemwingie combined to play in Morrison. This time the Nigerian's touch was inch perfect but his team-mate's was not, allowing Jaaskelainen to take the ball off his toe.
In the 68th minute, Odemwingie again found himself with only the keeper to beat and, although his effort was better this time, it was still a couple of yards the wrong side of the post.
Wanderers finally fashioned a chance of their own moments later as substitute Sam Ricketts crossed for Elmander, who planted an overhead kick onto the roof of the net from six yards.
Baggies boss Roberto Di Matteo sent Ishmael Miller on for Morrison, and the striker almost had an immediate impact only for Cahill to somehow get in the way of his goalbound shot.
The England defender was then involved at the other end, first seeing a shot blocked by Gabriel Tamas and then winning a free-kick just outside the box when he was fouled by Dorrans.
Stuart Holden stepped up and his 20-yard effort looked to be curling in before Carson flung himself to his right to tip the ball round the post.
Lee, who now heads off to the Asian Cup, received a standing ovation as he was replaced by Ivan Klasnic and, with four minutes left, the home fans were cheering again when Elmander made the points safe.
After breaking away down the right, the striker tried to play in Klasnic. But, when the ball came back to him, he took it on himself and slotted a shot under Carson from the tightest of angles.
Captain Kevin Davies then tried his luck from the edge of the area after beating Carson to a long ball but was denied by Paul Scharner on the line.
Five minutes of injury time came and went before Bolton were able to celebrate their first clean sheet since the opening day of the season.