Coleman decided to pick Eastwood and name Morrison among the substitutes against Posh at the Ricoh Arena as his team searched for their first home win since September.
"I am delighted for everybody that we have got three points," Coleman said.
"We had to fight hard for it. At 2-2 we could have crumbled but we didn't and for Freddy to get the winner as well was fantastic.
"This was harder than Wednesday's game against Newcastle. We were 2-0 up at half-time but we know it is never easy at Coventry City. Something always happens and it did.
"Fair play to Peterborough, they forced us back in the second half. When they got the first goal back, you could smell the fear was back.
"I thought we showed great character to come back and win because there was a lot of pressure on everybody.
"Clinton headed one off the line which is just as important as scoring.
"He hasn't scored a goal for a while and Clinton being Clinton he is affected by that. He is very serious about his football.
"Clinton wasn't dropped today - I just wanted to take him out of the firing line.
"He said, 'You just watch Freddy, you put him up top and he will score a hat-trick.' Maybe Clinton is a better manager than me I don't know."
The Sky Blues started poorly against the team at the foot of the Championship table as Peterborough dominated the opening exchanges.
Chris Whelpdale pulled the ball back into the path of Lee Frecklington and the midfielder's rising shot from 10 yards arched on to the roof of the net.
Mark Cooper's men - who have yet to win since the former Kettering boss was brought in five matches ago - created another chance when Paul Coutts set up George Boyd. The left winger unleashed a shot which Keiren Westwood palmed away.
But the match turned on its head when Eastwood netted twice in two minutes.
Bell's corner from the right was nodded down by Aron Gunnarsson and Eastwood knocked the ball in from almost on the goalline for his second goal of the season in the 33rd minute.
He did not have long to wait for his third goal of the campaign. A long throw from the right from Gunnarsson was flicked on by Leon Barnett to Eastwood and he swiftly nodded the ball beyond Joe Lewis.
With their confidence restored, the Sky Blues went in search of further goals. Gunnarsson tested the keeper with a low shot from outside the box and Leon Best struck a firm shot which Lewis turned away at his near post.
Peterborough had the first real chance of the second half when a massive throw from Exodus Geohaghon caused problems in Coventry's defence.
The ball dropped to the unmarked Aaron McLean and the striker blazed a close-range shot high into the stand from eight yards.
And it was a long throw in the 66th minute which led to Peterborough pulling a goal back.
Geohaghon threw the ball into the box and Craig Mackail-Smith popped up to slam in a close-range shot.
The Posh, who have yet to win away in the league this season, equalised four minutes later. Westwood failed to hold on to a long-range shot from Boyd and Mackail-Smith hit the rebound past the keeper.
Coleman's troops went back in front in the 72nd minute.
The hosts were awarded a free-kick following a foul on Gunnarsson. McIndoe floated the free-kick to Best and the leading scorer headed the ball on to Eastwood. He fired a half-volley into the top corner from 10 yards for his first hat-trick for Coventry.
Coventry survived a late scare when the visitors could have equalised in injury time but Mackail-Smith's header bounced wide of the far post and then Morrison's late, late clearing header secured the points in the home team's favour.
Disappointed Peterborough manager Cooper said: "I could quite easily be sitting here with three points. I'm not, I'm sitting here with nothing mainly because we didn't really get after Coventry.
"We didn't defend set pieces. The three goals we have conceded came from set pieces and if you do that you're not going to get anything.
"It is people not doing their jobs - individual errors. Everybody has a man to mark and if you're marking that man and he scores a hat-trick then it is your fault I'm afraid.
"We are seven points adrift but we're not giving up. We will keep going. We're far from the worst team in the league."