The Sky Blues showed two sides to their game to overcome the npower League One leaders. They dominated Ronnie Moore's men in the opening period and should have had more to show for it than Leon Clarke's 19th-minute strike from close range.
But Rovers came back at City in the second period and after being in complete control in the opening period, they ended up hanging on in the second half. Tranmere came close to drawing level - first in a goalmouth scramble - then when goalkeeper Joe Murphy produced a wonderful reflex save to keep out Zoumana Bakayogo's header.
Robins said: "They showed winning qualities. I thought we played really well in the first half. We played less well in the second half.
"They changed their shape for the first time this season which was a compliment to us and we showed that they warranted doing that in the first half because I thought we passed it really well. We deserved the lead. All the stats were in our favour in the first half and in the second half I thought they edged it.
"We started to become a little bit sloppy and they caught us on the break and outnumbered us a couple of times and it took an inspired Joe Murphy to get us out of the mire and keep a clean sheet. The important thing was the three points and the players have come up trumps so I am delighted on two fronts."
Rovers manager Moore said: "I think without doubt we should have got a point. I thought we were poor in the first half. I thought Coventry bossed it.
"We tried to change it a little bit with three players up front who can catch pigeons. We never got going but then credit to them in the second half, we gave it a go.
"One of our bigger lads got done with the header at the corner and Clarke did what he is good at and scored from the knock-back.
"It was always going to be a tough game. When you look at it, Tranmere are coming here and are disappointed we have lost. When we got the ball we didn't exploit their lack of pace at the back in the first half. We played better in the second half when we went back to 4-4-2 but we need to be adaptable."