The Baggies find themselves perched just above the drop zone in 16th position, three points clear of danger, with five fixtures remaining.
With Albion next scheduled to meet the title-chasing Manchester City on Monday, captain Brunt is looking for improved performances so the West Midlands outfit can heap pressure on sides such as Aston Villa, Swansea City, Norwich, City, Fulham, Cardiff City and Sunderland who are all deep in the relegation mire.
"What we have to do is keep the pressure on everyone around us because the picture at the bottom changes so fast," he told Express & Star.
"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.
"We can't rely on other results. Our goal difference isn't too bad - that's a positive. It's in our hands."
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