Sheffield United was a man down early on after captain John Egan was sent off for a foul on Ollie Watkins.
Villa keeper Martinez saved John Lundstram's first half penalty in his first appearance after his £17 million ($21 million) move from Arsenal last week.
Konsa sealed the points in the second half with a looping header, a well-earned reward for a committed display after he played on despite appearing to injure himself sprinting out for the second half.
It was the perfect start for Dean Smith's side, which beat Burton Albion in the Carabao League Cup second round last week, as it looks to avoid a repeat of last season's relegation battle.
"A very good start, to get three points and a clean sheet you can't ask for more. I thought we started the game the better and the game changes with the sending off," Smith said.
"We probably should have scored more and since lockdown, defensively I think we've been more aggressive. It was a great start for Emi Martinez."
After losing at home to Wolves last weekend, United have now lost their last five league games dating back to last season.
"I wanted to clear up (with the referee) the decisions he made. Why wasn't there a look at the monitor?" United manager Chris Wilder said of Egan's dismissal.
"Both boys are grappling there. The linesman never once raises his flag for a foul. I don't see many Villa players running to the referee.
"It's Premier League football and you wonder what's going off."