Best and Mifsud had fired cash-strapped Coventry into a 2-0 lead after just 11 minutes before Albion launched a stunning second-half fightback.
Roman Bednar, with his third goal in three games, reduced the arrears in the 51st minute before Chris Brunt dragged Albion level 13 minutes later.
But the whole game swung on the 70th minute dismissal of Albion left-back Paul Robinson. He was shown his second yellow card for a foul on Mifsud, just seven minutes after being booked for kicking the ball away.
Coventry took full advantage of their extra manpower with Best reacting first in a crowded goalmouth to drill home a drive at the near post in the 82nd minute.
Best then turned provider six minutes later to set up Mifsud who walked the ball around Dean Kiely for his 13th goal of the season.
It was an extra sweet double for Mifsud who was back in action for the first time since being sent off against Albion last month when City lost 4-0 at the Ricoh Arena.
The win was also well timed for Coventry who are battling to avoid going into administration.
Their success also ended Albion's unbeaten home record in the league as Mowbray's team ran out of steam following a fourth game in nine days.
Albion faced an uphill battle after being stunned by Coventry's quickfire double - ironically only after smart third minute save from Dimi Konstantopoulos had prevented Zoltan Gera breaking the deadlock.
They fell behind just two minutes later after Bostjan Cesar failed to deal with a long ball from Ben Turner.
Best latched on to the pass and muscled his way into the penalty area before lobbing the ball over the advancing Kiely and into an empty net.
Albion were still reeling from that setback when Mifsud took advantage of being in acres of space to pounce inside the six-yard box.
Leon McKenzie surged past right-back Carl Hoefkens after being picked out by Jay Tabb and crossed for Mifsud who had the simple task of steering in a close-range header.
Coventry almost went further ahead two minutes later when Michael Doyle flashed a shot just wide.
Predictably Albion launched a second-half fightback with Bednar heading home a 51st minute cross from Paul Robinson.
Brunt, with his second goal since his £3million summer switch from Sheffield Wednesday, then completed the fightback with an inswinging shot from 20 yards
But Robinson's rash bookings then proved the key as Coventry powered on to collect only their fifth win in 13 games.