Neither side could find a breakthrough in a hard-fought but largely unexciting 90 minutes in the West Midlands.
The hosts began brightly as they created three decent chances in the opening 20 minutes, but none of them was converted.
The visitors could not impose themselves on Albion, who wasted yet another good opportunity when Rondon ran onto Brunt's measured pass but sliced a left-footed effort wide.
The result means that West Brom's winless run now stretches to 18 Premier League matches and Alan Pardew still awaits his first win as Baggies boss.
West Brom (4-4-2): Foster; Dawson, Hegazi, Evans, Gibbs; Phillips (McClean 88'), Livermore (Krychowiak 69'), Barry, Brunt; Rodriguez, Rondon (Robson-Kanu 53').
Everton (3-4-3): Pickford; Keane, Holgate, Williams; Kenny, Schneiderlin, Davies (Baningime 88'), Martina; Bolasie (Lennon 61'), Calvert-Lewin (Niasse 61'), Sigurdsson.body check tags ::