A point against the current league leaders was enough to lift the Baggies out of the Premier League drop zone, but means they are now without a win in six.
Brunt concedes that positive results need to be picked up if Pepe Mel's men are to avoid becoming locked in a basement battle, but sees plenty of positives heading forward.
"There are good draws and bad draws, and that's a good draw," he told Sky Sports.
"But our league position shows that we've had too many draws this season. We need to start turning those into wins very quickly but, against the top side in the league, that's a good point for us."
Asked if the players are concerned by the club's lowly league standing, Brunt added: "Massively. You can't deny it, we're in fighting with probably 10 other teams - which is good that there are plenty of teams down there for us to claw our way out, but we can only affect what goes on here and in our games.
"We're up for the challenge and hopefully tonight is the first step on the road to recovery for us."
Brunt believes West Brom showed what they are all about against Chelsea, with a collective effort allowing them to contain a title-challenging outfit for long periods.
He said: "We had a game plan, we stuck to it, the lads worked their socks off and to concede a goal from a set-piece before half-time is disappointing.
"We said about staying in the game - in the last few games we've given away goals in the first half. Credit to all the boys, we grafted away in the second half and as the game wore on there was only one team winning that, and I think Chelsea will know they've had a good game tonight.
"Our home games are going to be important for us for the rest of the season, we've got some big ones coming up. All we can do is keep together and keep plugging away.
"We showed tonight what sort of spirit we've got between the lads and the substitutes when they came on had a big impact, so it's great for everybody to be involved. We've got a 10-day break now so there is a lot of work to do on the training ground."