The Hawthorns outfit sit three points clear of the relegation zone, with a game in hand over the next three teams below them.
Brunt made his goalscoring comeback from injury in Saturday's 3-3 draw against Tottenham, and feels that the return to fitness of a number of other squad members will prove beneficial in the side's remaining five fixtures.
"It's been really tough having to sit games out in the last few weeks, just as it was for the likes of Billy (Jones) and a few others," he is quoted as saying by The Birmingham Mail.
"With players starting to come back for these last few games, hopefully the competition for places and the ability to freshen things up will help us get the results that we need to finish the job and preserve our Premier League status.
"What we have to do is keep the pressure on everyone around us because the picture at the bottom changes so fast.
"A couple of weeks ago it looked all over for Fulham, but their win at Villa has given them a chance and dragged Villa right back into it, just as we pulled Norwich into the scrap.
"All we can do is look after ourselves, look after our performances and try to get a couple more wins as quickly as we can."
West Brom visit title contenders Manchester City in their next Premier League fixture on Monday.
‘He instantly popped into our ratings as one of the best U21 midfielders in Europe. The impressive thing is Slot has made him into more of a no.6' Inside Ryan Gravenberch's transformation at Liverpool
‘I don’t think Liverpool would look at Ollie Watkins, a striker isn’t a pressing issue for them – it’s Arsenal who need one’ Former Reds star explains why his old club don’t need an out-and-out forward this summer
