The Baggies hosted their promotion rivals at The Hawthorns knowing seven points from their remaining three games would be enough to secure a return to English football’s tier.
And they took another step towards achieving their goal despite being held by Scott Parker’s side.
The Cottagers, who have already confirmed a play-off place, also entered the match with hopes of finishing in the top two.
And both teams had chances to win it with Grady Diangana forcing Marek Rodak into an excellent stop while Anthony Knockaert rattled the bar for Fulham.
Overall, though, this was very much a clash where two good teams cancelled each other out.
The important thing for Albion, though, is that their fate very much remains in their own hands with wins over Huddersfield on Friday and QPR next Wednesday guaranteeing they will go up automatically.
Despite largely being pleased with Albion’s display at Blackburn, Bilic opted to tweak both his team and system for the visit of Scott Parker’s side.
Opting for a 4-2-3-1 formation, both Ahmed Hegazi and Grady Diangana earned recalls to the starting XI with Kyle Bartley and Filip Krovinovic the men to make way.
It meant the Baggies lined up with Diangana on the right wing, Kamil Grosicki on the left and Matheus Pereira in the number 10 role. Charlie Austin continued as the lone striker.
Fulham lined up in a 4-3-3 system with Aleksandar Mitrovic – the Championship’s joint top scorer – leading the line and Ivan Cavaleiro and Anthony Knockaert either side.
The game plans of both teams were evident from the first whistle with Fulham – with the extra man in midfield – determined to dominate possession.
Albion, meanwhile, were content to sit back, set traps and try and hit Parker’s side on the counter through the pace of Grosicki and Diangana.
It was Fulham who controlled the opening 20 minutes, though, with Parker’s side moving the ball well and creating two decent chances to open the scoring.
First Knockaert left fly with a curling strike from 25-yards.
Replays later showed the effort was going wide but a diving Sam Johnstone couldn’t be sure and did well to tip the ball away.
A deep cross to the back post was then met by Cavaleiro who did well to steer the ball towards goal.
But Hegazi – who had impressed from the off – was perfectly positioned to hack the ball clear from just in front of his own goal-line.
Following a drinks break, Albion found some rhythm and started showing more aggression with Marek Rodak having to be alert to keep out a miss-hit cross from Dara O’Shea.
The Baggies then created their best chance of the half when Austin did well to hold off a defender before standing up a cross to the back post.
That delivery was met by Pereira but, on the volley, the Brazilian fired wide when he really should have hit the target.
For the rest of the half the game was very much like a game of chess – with two good sides cancelling each other out.
But Fulham did threaten again just seconds before the interval with Mitrovic meeting a corner only to see his header cleared from under his own crossbar by Austin.
Bilic opted to make a change at the break with Matt Phillips replacing Grosicki.
But things remained cagey as the clock ticked towards the hour mark with Fulham squeezing play up the pitch with a high defensive line and Albion lacking the quality to play their way through.
To give his team a different kind of threat, Bilic introduced Hal Robson-Kanu for Austin in the 65th minute.
And within moments, the Baggies had created two glorious chances to open the scoring.
First a cross from O’Shea was flicked on by Robson-Kanu to Diangana at the back post.
The winger brilliantly cut inside his marker to have the goal at his mercy. But his low drive was then brilliantly kept out by the outstretched leg of Rodak.
Then from the resultant corner, Pereira whipped in a cross which Semi Ajayi headed over.
Suddenly the game had become open with Fulham a whisker away from breaking the deadlock when a long ball forward was knocked by Mitrovic for Knockaert.
The Frenchman then launched a dipping and swerving strike from 20 yards which rattled the top of the crossbar before flying over.
With Fulham needing to win to realistically keep alive any hopes of winning promotion automatically, the game became a lot more stretched in the closing stages.
Albion had a half chance with Harrison Reed making an excellent challenge to stop Robson-Kanu getting a shot away inside the box.
The Cottagers then pressed hard in the closing minutes and forced a number of corners – but led by outstanding Hegazi Albion were able to clear.
And that meant in the end both teams had to settle for a point – with Albion knowing if they win their final two games they will secure promotion to the Premier League.
Albion (4-2-3-1): Johnstone, O’Shea, Hegazi, Ajayi, Townsend, Sawyers, Livermore, Diangana (Robinson 85), Pereira (Krovinovic 90), Grosicki (Phillips 46), Austin (Robson-Kanu 66).
Subs not used: Bond, Furlong, Bartley, Harper, Brunt..
Fulham (4-2-3-1): Rodak, Odoi, Ream (Christie 90), Hector, Bryan, Reid, Reed, Knockaert, Onomah (Johansen 77), Cavaleiro (Kebano 43), Mitrovic.
Subs not used: Bettinelli, Mawson, McDonald, Arter, Le Marchand, Sessegnon.
Referee: Geoff Eltingham