The Light Blues gaffer had come under severe criticism from fans in recent days after the champions' Ramsdens Cup final defeat to Raith Rovers at Easter Road last weekend, with some wanting him sacked.
Rangers have turned in a series of below-par performances on their way to the League One title this term but upped their game against United as they came close to a semi-final shock at Ibrox.
Missed chances and lapses came back to haunt the Light Blues as Jackie McNamara's side ran out 3-1 winners to reach the Parkhead final.
But Dodds, who had four years at Ibrox and lifted the Scottish Cup with the Gers in 2000, insists McCoist deserves credit after his side raised their game on Saturday.
He said: "He got it spot-on at the weekend. Jon Daly won a lot of balls in the air and, even when he didn't win it, he caused United problems and that is how Dean Shiels got his chance, which he should have done better and stuck it away.
"Big Daly caused United problems all day but it is limited, there is not much mobility from him and you have to create chances for him to score goals.
"I think Ally got his tactics right and he has answered a few of the questions put at him this week building up to the United game."