Making his 525th Premier League appearance, Speed beat Bryn Gunnarsson to the ball in the 31st minute when El-Hadji Diouf delivered the corner and found the net with a firm header.
France striker Anelka extended Bolton's lead in the 55th minute. Released by Diouf, he managed to evade a weak challenge from Ulises de la Cruz to score his third goal in four games.
Braaten added a third for Bolton in stoppage time with a clinical finish following a superb cross from Anelka.
It was his first goal for the club since arriving from Rosenborg in the summer and he savoured the moment.
Lee said: "I was pleased to get the victory as I have said all along we needed to be patient.
"But I am not going to get carried away. That would be wrong as one success does not cancel out three defeats.
"For me it is all about the players and the way they responded to those setbacks has been superb.
"Clean sheets are vital to any team and that was the most pleasing aspect of our performance."