The Baggies - looking for their first win since returning to the Premier League - were 3-0 up at the break thanks to two Callum Robinson goals and a clever Kyle Bartley strike.
Chelsea threw the kitchen sink at them throughout the second half and pulled one back through Mason Mount in the 55th minute.
Albion, though, did well to weather the storm and were hugely unfortunate when Callum Hudson-Odoi reduced Chelsea's arrears further.
In the build-up to the goal, it looked like there had been a foul on Darnell Furlong.
But the referee allowed play to go on with Hudson-Odoi firing in following a one-two with Kai Havertz.
Again Albion dug deep and did everything they could to keep Frank Lampard's side at bay.
But in the third minute of added time, Chelsea won a succession of corners.
Mount then saw a low drive parried by Sam Johnstone with Abraham tapping in.
Bilic reiterated prior to kick-off that he'd been pleased with the performance of his team for an hour against Leicester and up until Kieran Gibbs' red card at Everton.
Eager to build on those positives, the boss kept faith with the 3-4-3 formation used in those games - a system that allows the Baggies to switch to a back five without the ball.
With Gibbs suspended, Conor Townsend made his first Premier League start at left wing-back.
Furlong was on the opposite flank with Romaine Sawyers partnering Jake Livermore in the middle of the park.
In attack, Robinson was the central striker with Grady Diangana on the left and Matheus Pereira on the right.
Chelsea lined up in narrow 4-2-2-2 formation.
Timo Werner and Abraham were the front two with Mount and Havertz operating just behind.
In goal, Willy Caballero was given the nod ahead of the error-prone Kepa Arrizabalaga with Lampard opting to wait before handing a debut to Edouard Mendy who arrived from Rennes earlier this week.
In what was a truly remarkable opening half-hour, it was Albion who opened the scoring after just four minutes.
Pereira fed Robinson who entered the box before shifting the ball onto his left foot.
And the former Sheffield United man then arrowed a spectacular strike across goal and into the bottom corner.
Chelsea appeared responded to the setback well.
Reece James whipped ina fabulous cross that Abraham side-footed wide when he should have scored.
Werner then hit the crossbar after meeting a low Mount delivery.
It felt like Chelsea's pressure would eventually tell.
But in the 25th minute, Albion doubled their advantage.
Making his debut Thiago Silva, dallied on the ball with Robinson nipping in to win the ball and charge down on goal.
And yet again the Albion forward was clinical with Robinson hitting another low drive which flew past Caballero.
Just two minute later, the Baggies were left pinching themselves when Bartley made it 3-0 with a corner routine straight from the training ground.
Pereira's delivery was brilliantly flicked to the back stick by Furlong.
And there waiting was Bartley who kept his composure to fire in from close range.
Chelsea went on to dominate the ball for the rest of the half - with Lampard's side reaching the break having had 77 per cent possession.
But orchestrated by Bilic, Albion's organisation and discipline had been fantastic - with the Baggies defending deep to keep the visitors at bay.
Chelsea made a double change at the break with Hudson-Odoi and Cesar Azpilicueta replacing Kovacic and Marcus Alonso.
It meant a change in system with Hudson-Odoi playing wide right, Werner drifting to the left with Abraham up top.
Chelsea continued to dominate the ball following the re-start with Albion defending with a bank of four in front of a bank of five.
And they pulled a goal back in the 55th minute when Mount spun Livermore 25-yards out before arrowing a strike into the bottom corner.
With Albion penned in, Bilic made his first change in the 65th minute with Hal Robson-Kanu replacing Robinson.
But in the 70th minute, Chelsea had a second.
It looked as though Furlong was fouled in the build-up. But Chelsea then broke with Havertz and Hudson-Odoi playing a one-two in the box before the England winger fired in.
Albion continued to dig in from that point and restricted Chelsea to very little.
But deep into five minutes of added on time, Chelsea snatched a point when - following a corner - Mount saw a low drive parried by Johnstone.
Abraham, though, was in the right place to tap in from close range - with Albion believing their had been a handball by Havertz in the build-up to the goal.
Albion (3-4-3): Johnstone, O'Shea, Bartley, Ajayi, Furlong, Livermore, Sawyers, Townsend, Pereira (Field 90), Diangana (Phillips 74), Robinson (Robson-Kanu 65).
Subs not used: Button, Kipre, Harper, Edwards.
Chelsea (4-2-2-2): Caballero, Alonso (Azpilicueta 46), Silva (Giroud 72), Christensen, James, Kanté, Kovacic (Hudson-Odoi 46), Havertz, Mount, Abraham, Werner.
Subs not used: Kepa, Tomori, Jorginho, Barkley.