The Manchester United legend had an ill-fated 31-day spell in charge of fellow fourth-tier club Oldham Athletic last year.
He is stepping in on a caretaker basis after Salford, unbeaten in the league this season, decided to terminate Alexander's contract after a reign that started in May 2018 and included promotion from the Conference National.
Scholes' first game at the helm of Salford, who surrendered a two-goal lead to draw 2-2 with Tranmere Rovers on Saturday, will come at Port Vale this weekend.
The 45-year-old is one of the part-owners of Salford, with former United team-mates David Beckham, Ryan Giggs, Nicky Butt and Gary and Phil Neville also invested in the club.
Former Dons player Louie Barry scored his fifth goal of the season for the visitors before the home side equalised through Mo Eisa.
Stockport went back in front on the stroke of half-time though through Nick Powell and this time there was no way back for the hosts.
Alexander said: “The first half cost us. There was no intensity, no speed or no commitment to what we’ve been doing of late.
“We can’t lie to anybody and think we deserved anything from that game today.
“The way we started was really slow, took too many touches and delayed everything.
“If you don’t commit as hard as you have to in professional football, you get beat.
“The game was there for us as a team after last week’s performance, but we just didn’t get going in the first half.
“In the second half, there was an increase in intensity, but we didn’t trouble the goalkeeper or their goal enough to get anything from the game. The right team won today.
“I’m raw because I’m frustrated that a group of players who have done so well so far this season have not got anywhere near the levels we have done in previous games.”
Stockport manager Dave Challinor praised his players as built on their 2-1 win over AFC Wimbledon.
The two wins have followed a sticky start to the season, leaving Challinor delighted with the way his side have bounced back.
“We should take more confidence from the fact of how we went about it,” said Challinor
“Two big wins, especially on the back of what we’d had previously and especially on the back of being a little bit scarred, I suppose, by some of the games earlier on.
“To get those confidence-boosting wins like we have is massively important going into a massive game next week.
“The easiest thing for us today would be potentially to have gone 3-5-2 to match the systems and it becomes who’s better at the system. But we thought we can tweak the system in terms of pushing Powell higher up to become a 4-3-3.
“I’d like to think we can be adaptable. We’ve got the personnel to play as a four and as a three, but it’s ultimately about winning football matches and we’ve won the last two in fractionally different types of that formation, asking different questions of the opposition.”
