James Morrison's side went in front through Aune Heggebo's 16th-minute goal at Pride Park.
Albion's Brayann Pereira was sent off on his debut for a second yellow card in the 34th minute.
The visitors were under almost constant second-half pressure but Derby created few clear-cut chances, with the best falling to striker Lars-Jorgen Salvesen, who headed over from eight yards.
West Brom's fourth win from six league games took them one point clear of second-placed West Ham, who won at Bolton on Tuesday.
Fourth-placed QPR missed the chance to go top of the table as they were held to a 0-0 draw at fifth-placed Charlton.
The Hoops knew a third straight away victory would take them to the summit, but they were unable to rise to the challenge in a tepid London derby.
Norwich, in ninth place, beat Birmingham 2-1 at Carrow Road to make it three wins in four league games.
First-half goals from Oscar Schwartau and Papa Diallo saw off Birmingham, whose second-half effort from Carlos Vicente was scant consolation.
The Spaniard wasted a glorious chance to equalise near the end as Chris Davies's men slipped to their first league defeat of the season.