Bullard stepped off the substitutes' bench and struck in the 93rd minute in his first appearance since early October following knee ligament damage.
Tigers striker Jay Simpson followed up his first two goals for Hull last week by firing a first-half double before the Blades, booed off by a section of their fans at half-time, launched a brave fightback after the interval to haul themselves level through two Ched Evans goals.
Carver, taking charge of his second game following Gary Speed's departure to become Wales coach, said: "I love coaching and working with the players and nothing has changed from my point of view."
The 45-year-old added: "The fact that we conceded a goal so early was disappointing because it wasn't as if we hadn't prepared for what they were going to do. We knew that they were going to get the ball behind us and turn us.
"I thought in the second half we looked like the side that was going to win the game. I'm gutted for the lads because they gave everything. That had a right go and dragged themselves back into the game."
Source: PA
Source: PA