Noah Okafor struck twice inside the opening 30 minutes to spark wild scenes among the travelling fans, before Martinez was sent off 11 minutes into the second period by referee Paul Tierney after a VAR intervention for tugging on Dominic Calvert-Lewin's hair.
United's 10 men rallied after the break and pulled one back through Casemiro's 59th-minute header, but Leeds held on for their first league win at Old Trafford in 45 years.
"He goes to grab his shirt and ends up touching his hair and the bobble comes out and you end up getting a red card for it," United's interim head coach Carrick said.
"It's not aggressive, there's no jolt, there's no tug, no sudden movement.
"We've got to be careful where the game's going if we're giving red cards for that ... It's a shocking decision, an absolutely shocking decision."
United lost at home for the first time under Carrick after five straight wins and he also felt Okafor's opening goal should have been disallowed for Calvert-Lewin's challenge on Leny Yoro in the build-up.
"I don't know how that doesn't get overturned," Carrick added.
"I have to say, the goal itself, Leny Yoro gets a forearm smash on the back of the head, which is worth looking at again.
"What are we getting sending-offs for? It's a clear forearm smash on the back of his head."
Leeds climbed six points clear of the relegation zone after their second away win of the season and their first in the Premier League at Old Trafford.
The West Yorkshire club have not won in the league at their fiercest rivals since 1981, while it was their first victory against United since their FA Cup win there in 2010.
"It feels outstanding," a delighted Leeds boss Daniel Farke said.
"It's 'crunch-time' period, we're playing a bit under pressure.
"We missed too many of our chances, could have buried the game before. We had to dig in, but no question we deserved to win the game."
Leeds face bottom-of-the-table Wolves at Elland Road on Saturday when they can climb nine points clear of the bottom three.
"It feels like under pressure we really deliver," the German added.
"Nothing is achieved yet, I'm not over confident or celebrating.
"We're proud of the win, but don't forget where we're coming from. We have to stay on it. I don't want to celebrate too much."
