An own goal from Gary Thom in the early stages of the second half set the visitors on their way before Irish striker Jon Daly claimed a hat-trick.
McCoist said: "It was comfortable after an hour. I wouldn't have said it was comfortable at half-time. I thought we did okay in the first half, no more than okay.
"I thought we lacked a bit of urgency in our forward play. We were tidy enough without being aggressive in terms of attempting to score a goal.
"That certainly changed in the second half when I thought we got our just rewards."
East Fife manager Willie Aitchison admits his team ran out of steam after the interval against the League One leaders.
"We did well for 45 minutes, but games last for 90 minutes," Aitchison said. "We kept our shape in the first half and worked hard to contain Rangers and it worked.
"However, the over-stretch for the own-goal did not help the guys' confidence. I doubt very much any team in the league will stop Rangers before they win the title.
"You're talking about a team that will have a win rate this season of 85-90 per cent. They have 45 times our budget. That's what you're up against."