On-loan Everton midfielder Dowell, who moved to Bramall Lane in early January, struck after 14 minutes to move United into second place on goal difference ahead of Leeds – who have a game in hand – and two points behind leaders Norwich.
Albion had been chasing a fourth win in five games to strengthen their own bid to reach the Premier League, but the defeat means they are four points adrift of the automatic promotion places and a Blades side that has now won three in a row without conceding a goal and are unbeaten in their last six.
Albion had been boosted by the return of 16-goal striker Dwight Gayle following the completion of a two-match ban but even with the on-loan Newcastle marksman back in the side they failed to find a way through a well-drilled United defence.
Albion did have the ball in the net six minutes from time but it was rightly ruled out by referee Oliver Langford as it flew in off the hand of Jay Rodriguez when he tried to head home Mason Holgate's cross.
Rodriguez was quickly into the action at the start of the game as within the first five minutes he fired a shot over from 20 yards before seeing a header flash over the bar.
It proved to be a false dawn for Albion as United made their first real chance pay.
When Kieron Freeman surged down the right-wing his ball through to Martin Cranie looked set to roll behind for a goal kick.
Cranie, however, managed to wrap his foot around the ball and hook a cross into the centre of the area where Dowell powered home a header that flew beyond goalkeeper Sam Johnstone.
While Johnstone was powerless to stop Dowell, he did prevent Albion falling further behind five minutes later with a stunning one-handed save to turn away a 20-yard drive from Marvin Johnson.
The home side then enjoyed a flurry as they went in search of an equaliser, with Holgate's curling shot dropping wide in the 24th minute before United goalkeeper Dean Henderson held on to a header from Craig Dawson.
With United proving hard to break down, it took Albion 20 minutes into the second-half to test them again.
Jake Livermore floated a ball to the far post that was met by Gayle on the volley, only for United full-back Enda Stevens to bravely block.
Albion then dominated in terms of possession but United showed they could still carry a threat when they ventured forward.
Skipper Billy Sharp was narrowly off target with a glancing header in the 77th minute before Gary Madine fired straight at Johnstone 10 minutes from time.
Henderson then denied Albion an injury-time equaliser with a save from Jefferson Montero.