Marvin Morgan's 62nd-minute strike was the difference between the two sides and also saw the Shrews complete the double over Coventry after their 4-1 win in the previous meeting.
Coventry went into the game unbeaten in 10 games and were left wondering how that run had come to an end after registering 21 efforts on goal.
But it was Shrewsbury who left with the points and Robins said: "We have made enough chances to win four or five games.
"We have wined and dined our supporters, but didn't see the job out. It's disappointing, we start the year in a negative way whereas we all wanted it to be positive, but we have to take it on the chin and try and move forward and build.
"They don't come any tougher than our next game (at Tottenham in the FA Cup), but it's a game where we can get this one out of our system and we can go and enjoy that experience and see how we do."
It was a bad day for Coventry, who were playing in front of their biggest home crowd of the season.
Gary McSheffrey wasted the clearest opening when he missed the target after a backpass landed in his path, but he was not the only one. David McGoldrick was playing the last game of his loan spell and went out on a low note.
"David McGoldrick had his poorest performance since I have been here," Robins added. "I wanted him to sign off in style, but you don't always get what you want.
"But he has been magnificent, he will go back to Forest and I will keep mithering the life out of him to see what his plans are, but whatever happens he has been fantastic for us."
- Sports & Recreation