Lee McCulloch scored his second successive hat-trick as five goals in the last half hour gave Rangers an emphatic win, if a rather flattering scoreline, against an Arbroath side which more than matched the Ibrox team for over an hour.
And McCoist freely admitted his side struggled to overcome their opponents until the later stages of the game.
The Rangers boss was delighted with the final outcome as his team netted five second-half goals after falling a goal behind to a very well-organised Arbroath.
And although the focus was on the home side, significantly boosted by the inclusion of six of McCoist's summer signings for the first time, the Rangers boss commendably praised his opponent's performance and the fact the scoreline was a bit generous in his side's favour.
He said: "It was a great win but I don't think anybody would have predicted that at half-time. As delighted as I am about our win and the manner it was achieved I have plenty of praise for our opponents.
"We knew they were a good team and it is obvious how they managed to get a 1-1 draw at Celtic last season and they are probably shattered and wondering what happened as they played really well."
McCoist was also keen to point out how difficult it was for all of his new signings in their first game together.
" I never said it would be easy throwing six of them into the game but I thought we started quite well but then went a bit flat. But credit Arbroath who knocked the ball about well and got in behind us a few times in the first half.
"After they scored, and I thought it was a terrific goal, we responded well.
"I thought the scoreline flattered us a bit but nobody would deny us the victory."
And as the club's leading striker, McCoist singled out McCulloch's second consecutive hat-trick, which put Rangers in control, adding: " Two back to back hat-tricks from a defender and without any penalties is fantastic.
"And the thing about big Lee, although he was a striker, he is delighted just to play anywhere in the team".
Arbroath player-boss Paul Sheerin was a bit disappointed by the final scoreline as he thought his side did well until Rangers grabbed the lead after 65 minutes.
The Gayfield boss was also honest in highlighting that his side failed to keep it going for 90 minutes.
Sheerin said: "The scoreline is possibly a bit harsh but the third goal, which was an unbelievable strike has killed us.
"After that we were forced to change things and that left us open.
"It was a great goal an finish from Cookie to give us the lead and I thought we did really well with one man up front for 65 minutes. The boys worked really hard and I suppose in that respect the scoreline is slightly harsh on their efforts.
"But there is no getting away from the fact that we lost 5-1 and the bottom line is we didn't compete for 90 minutes. And you have to credit Rangers for keep going and not giving up.
"Our game plan was to frustrate them, which we did for long spells until they scored their second goal."