Dobie nodded the ball past keeper Alan Combe to snatch three priceless points for the Baggies and stretched their unbeaten league and cup run to 10 games.
But, Bradford City can consider themselves extremely unlucky not to take something from the game after they created most of the best chances.
The Bantams almost got off to a dream start inside the first minute when Gareth Farrelly swept a teasing free-kick in the box from the left.
With keeper Russell Hoult off his line, Jason Gavin rose above the defence to meet the ball with a firm header. But, unluckily for him, his effort from eight yards bounced off the top of the bar and went out of play.
If that rattled West Brom it certainly worked, within minutes they had City on the back foot and could have opened the scoring themselves when Paul Robinson struck a fierce drive from a tight angle which whistled across the face of the goal and beyond the far post.
Welsh international Jason Koumas then saw his long range effort blocked before it fell to Robert Hulse. But the tall striker did not get much power behind the ball and his shot on the turn rolled into Combe's arms.
After 15 minutes, Koumas again fancied his chances firing a free-kick from 35 yards. He struck it well enough, but Combe had it covered as the ball drifted past his left hand post.
City were by no means phased and played some neat flowing football that belied their lowly position near the foot of the table.
The home side sprayed the ball around with confidence and self belief, Farrelly in particular was like a beacon light in the thickening gloom.
On the half hour mark City had claims for a penalty turned down when a flick by Dean Windass inside the box shot up and accidentally caught James O'Connor on the arm. Referee Mike Pike was right on the spot to wave away any protests and made the right decision.
Farrelly almost set up Windass with a hanging ball to the edge of the box. Windass ran towards it but Hoult sped off his line to snuff out the danger.
Three minutes before the break Farrelly again charged forward and laid off a fine pass to Andy Gray, but the attack fizzled out with a poor shot from the Scottish striker.
City could have broken the deadlock after 53 minutes when Nicky Summerbee's shot took a deflection off the referee. But Hoult always had his eye on the ball.
Scott Dobie should have scored for the visitors when Koumas set him up in the box. However the striker took too long to pick his spot allowing Combe to race out and make a desperate block.
The ball then cannoned off Paul Heckingbottom and rolled towards the net. But the defender spared his own blushes when he got back in the nick of time to clear.
West Brom boss Gary Megson decided to shake things up, replacing Phil Gilchrist with Lee Hughes who was roundly booed by the home fans as he stepped onto the pitch.
After 63 minutes, Windass almost poked City ahead when he raced to the back post to latch onto a low ball as it flew across the face of the goal, but Paul Robinson just did enough to beat him to it and cleared for a corner.
Seconds later, City came even closer to scoring when Heckingbottom's 25-yard drive was kicked off the line by Koumas.
Albion was battling to keep the Bantams at bay, but they had the last word only two minutes from time when Macedonian international Artim Sakiri came off the bench to deliver the cross that led to Dobie's goal.
The result was harsh on City as they did not deserve to lose, but they can't rely on hard luck stories to keep them up. But, under Bryan Robson, City can survive if they perform like this.
As for West Brom it will take something special to stop them returning to the Premiership.