The Blades, looking to repeat the heroics that took them to the semi-finals of both knockout competitions last season, were by far the better side in this FA Cup third-round clash.
But it was not until the introduction of former Huddersfield frontman Allison that they made their dominance tell and he headed home with 16 minutes left to settle the tie.
Having seen his side go five games without a victory, Cardiff boss Lennie Lawrence chose to shuffle his pack. Robert Earnshaw was promoted from the bench in place of Andy Campbell, while Alan Lee was named as a substitute at the expense of Chris Barker.
United also opted for new faces, with defender Chris Morgan returning, while new loan signing Dean Sturridge also started.
Striker Steve Kabba was not in the squad after breaking his leg last week against Wigan, but the Blades did not seem to miss his presence too much as they immediately forced the pace.
Virtually from the whistle they surged forward and after just 29 seconds they could have gone ahead. Following some neat passing in midfield, Phil Jagielka raced down the left and when he crossed, goalkeeper Martyn Margetson had to be alert to turn Jack Lester's shot round the post.
That was a clear statement of intent and for the rest of the half the visitors looked the more dangerous side.
After Morgan had seen a drive hacked clear by Spencer Prior, on 20 minutes United threatened again as Lester steered Sturridge's curling centre wide. Then, three minutes later, the same player fired close again when he latched onto Michael Tonge's free-kick.
Cardiff were being pinned on the back foot and it was not until just before the half-hour mark that they carved out their first chance when Tony Vidmar's header flew wide. That was the best they could muster, though, and in the end they were fortunate to go in level at the break.
After the restart they were soon under the cosh again. Nick Montgomery was beginning to dominate in the United midfield and after 52 minutes he surged forward before firing in a powerful shot. Then Stuart McCall unleashed a similar effort that also forced Margetson to save.
After 63 minutes the visitors' cause was hampered when Sturridge was forced off with an injury to be replaced by Paul Peschisolido.
At the same time Blades boss Neil Warnock sent on Allison in place of Ashley Ward and 11 minutes later that switch paid dividends.
The breakthrough came from Tonge's surge down the left and when he crossed into the box, Allison rose above his marker to power home a header and book the Blades a place in round four.