A section of the Carlisle crowd called for Abbott's head at the end of a game, which could easily have ended with greater embarrassment for the Cumbrians.
Carlisle have leaked 13 goals in their first three League One games and it was clear that the patience of their supporters was wearing thin as the home defence was cut apart by a rampant Sky Blues' attack.
But Pressley said: "Greg Abbott has been one of the longest-serving managers and has been through this sort of thing before.
"He will get this right, of that there's absolutely no doubt. He has been a fantastic manager of this club."
It was ironic that Coventry-born Abbott's home city club was responsible for inflicting this latest drubbing.
Carlisle made three changes to their defence, but all too often players were caught out of position and undone by pace by the lively Coventry frontmen.
Leon Clarke's 20-yarder and a solo effort by Franck Moussa set Coventry on their way inside a torrid opening six minutes for the home side and Billy Daniels picked his spot to make it 3-0 before half-time.
James Berrett missed a penalty for Carlisle and Clarke profited from Kevin Feely's mistake to add number four in the second period.
City should have had more but for a series of fine saves by Carlisle goalkeeper Mark Gillespie, while the woodwork robbed ex-Carlisle striker Mathieu Manset of a goal on his return.
Although Pressley expressed pride in a team effort, he had a special word for keeper Joe Murphy for his penalty save.
"There always are pivotal moments and Joe's save at 2-0 was impressive," he said.
"We look an exciting side, but the clean sheet was very pleasing."
"We are playing the sort of aggressive, fast football I have been wanting to see and the players deserve so much credit.
"Some of them were disappointed at half-time to be only 3-0 up. We showed energy and speed and the only disappointment was that we could have had a lot more goals."
Pressley added: "We were good, but we can be better.
"We were a bit nervous at the start of the second half, but once we got the fourth we were terrific for the remaining half-hour.
"In our predicament we have got to win games, but it was a step forward."
The win reduced Coventry's deficit to minus four points, while Carlisle are a place above them and still awaiting their first point of the season.
Abbott admitted he was "rock-bottom" and the performance marked the lowest point in his career.
But he pledged not to give in and said:"It's one of the toughest days I've had in football.
"It was embarrassing in the end and the dressing room is so fragile. Every little thing affects the players and I feel as a group we have let people down.
"The game was lost within five minutes, but we are better than that.
"The early goals killed our confidence. I'm not going to give in. I will take the bullets and fire some back when we deserve it.
"I'm not sure we have been as desperate as this in my time here.
"It has hurt like mad and if we don't put it right we will be relegated."