The home side needed just 16 seconds to open the scoring at Tannadice through Johnny Russell, who completed a double in the second half, after Jon Daly doubled the advantage, for a 3-0 win.
Mccoist said: "It was certainly poor. Probably our bigger SPL players should have done better today."
Rangers were without key players, such as influential skipper Lee McCulloch and youngster Lewis Macleod, for the trip to Tayside.
But they did have a starting line-up boasting players with recent experience of playing in the Clydesdale Bank Premier League - in the shape of Ian Black, David Templeton, Francisco Sandaza and Dean Shiels - and McCoist knows they failed to deliver on the day.
McCoist added: "I don't think it would be any good for me to be critical of individuals at this moment in time but I do take the point.
"It was a disappointing day for one or two of the boys who, with SPL experience, would hopefully have done better. When criticism is due and warranted, believe me, they get it. Some of the players got it today after the game."
The victory provided new United boss Jackie McNamara with the perfect start to his tenure, after replacing Peter Houston at the helm in midweek.
He said: "It was a great start to the game and I thought we carried a real threat and could have had a few more goals.
"We are delighted just to get through."
- Sports & Recreation
- Clydesdale Bank Premier League
- Lewis Macleod