The former Sunderland and Stoke midfielder was handed the temporary reins following the sacking of long-serving manager Greg Abbott during the week.
After hailing a "magnificent" response from his players, Kavanagh said: "Yes, I have applied for the job and yes, it definitely interests me.
"The players deserve a massive round of applause after what's been a difficult week.
"The last thing the gaffer said when he left the club was to ask the players to get a victory and that was the message, to go out and perform.
"One or two felt they had let him down, but they were magnificent and when we lost possession they closed the lines down.
"I have applied for the job, but it was not about me, but getting the first win for the club and the fans, who were top drawer and stuck with us at vital times."
Matty Robson's 54th-minute header was enough to secure Carlisle their first league win of the season. It came when the Blades' defence failed to deal with James Berrett's corner and although the ball only just crossed the goalline before it was scrambled away, the referee's assistant flagged for the goal.
The Blades weaved pretty patterns, but were punchless and the result left manager David Weir saying: "I'm disappointed and frustrated and questioning a lot of things."