Wells broke the 51-year-old record when he put Bradford in front a minute before half-time with his eighth goal in consecutive matches in front of a crowd of 18,041 - five this season and three in the final three games last season.
The Bermudian international took a headed pass from Garry Thompson and turned the ball past Neil Collins before scoring with a low right-footed shot into the far corner.
Then, after both teams had endured narrow escapes, Wells put the result beyond doubt with Bradford's second goal thee minutes from the end, turning the ball into the net from fellow striker James Hanson's pass after Blades goalkeeper George Long had made a superb save to deny substitute Raffaele De Vita.
The visitors' best chance came in the third minute when Jamie Murphy ran on to a through pass from Lyle Taylor, but Jon McLaughlin stopped his goalbound shot with his legs. The Bradford goalkeeper also did well to deny Febian Brandy and substitute Ryan Flynn.