Smith, who signed from Charlton this week, opened his account after just 30 minutes when he latched on to Yaser Kasim's cross to divert past visiting keeper Chris Weale.
Cooper, whose side climbed up to eighth in the table, said: "Michael is honest, he works his socks off and he can score a goal.
"I'm pleased he got off the mark so quickly and hopefully he can go from strength to strength.
"I'm not sure some of our other forwards would track back 60 yards. That's what I'm after - I want them to work hard for the football club.
"I'm pleased to get three points. It was an atrocious game of football, especially the way we tried to play.
"The most frustrating part for me was letting them score a goal just before half-time when they hadn't really threatened.
"We weren't pretty, but we got it done. I know the fans got frustrated.
"When you win every week at home sometimes it's going to be difficult to do so, but we got it done.
"There were too many silly mistakes, too many wrong decisions, but on days like this you just have to get three points and we did."
Shrewsbury, who lost manager Graham Turner in the week, scored a deserved equaliser through Parry after 44 minutes, the midfielder rifling in a free-kick after Raphael Bronco had brought down Jon Taylor.
After Asa Hall had hit the crossbar for Shrewsbury, substitute Ranger came off the bench to change the game.
Ranger rose highest after 89 minutes to head home from close range following Alex Pritchard's corner before going on to set up the third goal.
The former Newcastle player then set up Smith for the latter to complete his dream debut and make the points safe.
Shrewsbury caretaker-manager Mike Jackson, whose side are in the relegation zone, said: "It is a really cruel way to lose the game as we played so well in the second half.
"We did well to get back into the game and I thought we would go on and win it, but their substitutions changed the game.
"I am very proud of my players for their effort after a difficult week.
"They couldn't have given me any more and I am sure they will continue to do well.
"I've worked the last three and a half years coaching the team and the reserves.
"All I will say is that I'll do my very, very best to make sure I do what I'm good at."
- Sports & Recreation
- Nile Ranger
- Paul Parry
- Yaser Kasim