Coleman's men had hit the woodwork on four occasions - including the earlier Leon Best penalty miss - but the breakthrough came after referee Rob Shoebridge awarded a second spot-kick for a James Chambers push on substitute Gary Madine.
Coleman said: "It would have been a travesty if we didn't win. We were the better team.
"There were spells in the first half when Doncaster played some lovely football but we were patient and had good chances.
"That's our third win on the bounce and that's been thoroughly deserved because the players have been tremendous. They have stuck together.
"I was delighted for Clinton. It's not easy when you haven't scored for some time but as soon as it was given, I knew there was only one man who was going to take it."
Coventry's first chance came in the 13th minute when Patrick van Aanholt sent a cross into the box for Freddy Eastwood to head goalwards - but the post denied him a fifth goal in three games.
The next near miss could have sparked controversy. Play restarted after a head injury with an uncontested drop ball but Best's punt lobbed Neil Sullivan, only to hit the crossbar.
The pair were involved in the game's next major incident, when Sullivan brought down the Republic of Ireland striker it the box for a penalty, but Best's powerful shot rebounded off the right hand post and clear.
Doncaster had their best chance of the game in the 55th minute when an excellent run from Jon Oster into the City box caused problems, but when the ball eventually rolled to Dean Sheils eight yards from goal he blasted the chance over the bar.
Michael McIndoe struck the goal frame for a fourth time in the match with a 72nd minute volley, but after Coleman's gamble of bringing on Morrison and Madine, the 85th mnute spot-kick was hit home to give City the advantage.
Doncaster manager Sean O'Driscoll said: "I thought the first one was a penalty - I would have expected it if it was our striker. We got a bit of luck with the miss but the second penalty was a soft decision to say the least.
"I've seen it on the monitors and I have already clipped four or five incidents that are exactly the same. Why one is a penalty and another isn't, I don't understand. You just want a bit of consistency.
"I thought we were organised and disciplined and never really looked in danger of conceding apart from their long throws which are effective even if they are unattractive to watch."