The 60-year-old boss was making his managerial debut for the club having signed a two-year deal at the Stadium of Light during the international break.
His side are placed precariously in the Premier League table and headed to the Hawthorns in poor from with a defence which has been breached more times than any other top-flight side.
However, the Black Cats fared well until Saido Berahino’s second-half strike handed the hosts victory.
Indeed, the striker’s finish wasn’t without controversy as Sunderland goalkeeper Costel Pantilimon felt he was fouled in the process, an opinion Allardyce also shares.
"In our position that's a hugely precious point, even at this stage of the season. It's a big mistake by the referee,” the former West Ham United boss pointed out.
“It was a clear foul in two areas. One, he (Berahino) has jumped backwards into our goalkeeper when he is not even looking at the ball and two, he handled the ball in front of him to score."
Sunderland played in a positive manner and at times put the Baggies under pressure, but there was a clear lack of creativity in the final third and striking the right balance will be difficult.
Allardyce knows he can’t push too many men forward and leave a weak defence exposed, yet sitting back and becoming compact may starve his side’s forwards of any real opportunities.
Next weekend’s home clash against Newcastle United is very much a must-win game for Sunderland as their survival hopes will be pinned on picking up enough points at the Stadium of Light.