Martinez, signed from Arsenal last week, kept out John Lundstram's spot-kick in the 36th minute of the Villa Park clash on Monday night.
Ezri Konsa subsequently headed home just after the hour mark as the hosts made a winning start to their 2020-21 Premier League campaign.
Martinez said on avfc.co.uk: "When you get three points, a clean sheet and a penalty save; it's like scoring a hat-trick for me.
"I felt part of the squad since day one, that's credit to my team-mates and the staff.
"When you come to a new club and have only four or five training sessions, but the lads, fans and staff welcome you really nicely, it helped me to perform.
"I have to give confidence to the back four. That was my only target, I did it, and hopefully I keep doing that during the whole season, with consistency."
As well as signing Martinez, Ollie Watkins, Bertrand Traore and Matty Cash in the last few weeks, Villa have also announced new deals for England players Jack Grealish and – on Monday ahead of the Blades game – Tyrone Mings.
Mings has signed a contract extension running to 2024, and Villa boss Dean Smith said of the defender: "He's become a key member of the squad and the team. He's got a really good personality around the place.
"We're all happy at Aston Villa that he wanted to commit. Again, as I said after Jack – big statement of intent from our owners of where we want to go to."
Asked if he envisaged more players coming in before the transfer window closes, Smith said: "Possibly. We're still discussing. There's possibly the chance to get something, but we'll have to wait and see."
Monday's result made it two defeats from two league matches so far for Chris Wilder's Sheffield United, whose skipper John Egan was sent off in the 12th minute.
Egan was given a straight red card by referee Graham Scott, having tussled with Watkins and been judged to have denied the striker a goal-scoring opportunity.
Wilder felt there should not have been a red card on that occasion, and that one should have been shown to Villa's Matt Targett – instead of a yellow – when his tackle on Chris Basham resulted in the visitors being awarded their penalty. In both cases, the incidents had a VAR check.
Wilder claimed his team were "on the receiving end of two poor decisions", and questioned why Scott had not consulted the pitchside monitor.
Having said the match had left him "confused", Wilder added: "I can't give the solution, the referees and the officials have to find the solution.
"Why wasn't Graham being told to look at the monitor? Why was it so quick on the decision on the penalty? I don't understand. I don't mind (waiting), but get the right decision.
"The directive that the referees would go and use the monitor more frequently – certainly that hasn't happened (on Monday evening)."