Sammy Lee's battling side were just too good for Reading at the Reebok Stadium, with France international Anelka the driving force behind their impressive victory.
Speed scored the opener after 32 minutes, Anelka doubled the home side's advantage ten minutes after the restart and, when super substitute Braaten netted in injury time, Lee punched the air with delight after his side claimed three vital points.
As early as the ninth minute Anelka stamped his authority on proceedings.
The quick frontman beat his marker and fired a powerful right-footed shot goalwards only to see it blocked by Andre Bikey.
A minute later experienced Wales international Speed volleyed over the bar with a sweetly struck volley.
Emerse Fae saw a right-footed shot charged down by Jlloyd Samuel as Reading fought back, before Kevin Doyle headed agonisingly wide of the target in the 18th minute.
The visitors were now dominating proceedings and Shane Long saw a shot saved by Bolton keeper Jussi Jaaskelainen.
At the other end, Kevin Nolan should have done better when in a good goalscoring position, but he fired wide of the target.
But Bolton finally made the breakthrough when El Hadji Diouf's corner from the right was met sweetly by Speed and his header left Reading keeper Marcus Hahnemann no chance.
And Bolton came close to doubling their advantage three minutes before the break.
A burst of pace from Anelka allowed the striker to unleash a fierce right-foot shot, but Scott Golbourne did well to make a last-ditch tackle to deny the Frenchman.
Bolton started brightly in the second half and after seven minutes Diouf fired wide with a powerful right-foot shot.
At the other end, Ulises De la Cruz went close with a long-range effort but it didn't trouble Jaaskelainen.
However, Bolton took a major step to climbing off the bottom of the table when the impressive Anelka beat his marker and then beat goalkeeper Hahnemann with a rasping 20-yard drive.
Reading kept battling only for Hahnemann to be forced into action again. He saved well when Mikel Alonso broke free and hit the target.
In the 72nd minute Diouf saw a shot charged down by De la Cruz, before playmaker Nolan went close with a right-foot volley which flew inches wide of the target.
Anelka nearly bagged his second of the game three minutes before full-time but he again failed to hit the target when in a good position.
But victory was finally wrapped up when second-half substitute Braaten met a superb cross from Anelka and fired past Hahnemann to make sure of the points.