Matheus Pereira scored two penalties as the Baggies came from behind to claim a 3-2 win - just their second Premier League victory of the season.
It was also Allardyce’s first win since replacing Slaven Bilic as boss last month.
And he said: “This will give us a big boost. Not just us internally but also the fans, hopefully, can have a good weekend now we have given them something to cheer.
“It is the first Black Country Derby for a long time as we managed to clinch a victory and earn three very valuable points and only our second win of the season.”
Albion’s win was even more impressive considering they were without goalkeeper Sam Johnstone, their best player so far this season, after he tested positive for coronavirus.
Matt Phillips also missed out due to a positive test, while Grady Diangana was absent due to a hamstring injury.
The latter’s absence saw Kamil Grosicki make his first Premier League start since May 2017.
Allardyce said: “It is the first game Sam has missed in four years. I was comfortable in David Button coming in and replacing him as he did.
“Obviously Matt Phillips is very important to us. I thought Kamil did well, everyone did well.”