A second-half strike from Shane Long - with new assistant Republic of Ireland manager Liam Brady watching in the stands - and a late goal from substitute Dave Kitson proved enough for a second successive win for the Royals.
However, despite now climbing up to 13th place after a result which puts daylight between Reading and the drop zone, Coppell maintains there is still plenty of hard work ahead.
"We have to accumulate a certain number of points, and we cannot relax in any way," declared the Royals manager.
"We are all on a mission to ensure we are playing in this division again next season.
"It is congested down there.
"One week you can be out of the firing line, but we have to string a number of results together to make sure people are not looking at us under the microscope."
Coppell added: "It is accepted that 40 points is a significant number and I maintain we are capable of getting that, so that is my target."
The Reading boss was quick to praise the efforts of his front men.
"The example of Kevin Doyle and Shaun Long in terms of massive desire and effort, combined with a great deal of skill, to create the first goal made life a lot easier for us," Coppell said.
"However, for City to lose Richard Dunne was significant, because he is a fine player and a good leader.
"That played to our advantage, but we still had to make use of that advantage."