Robins, who won the Cup as player with United in 1990, watched his side battle back from 3-0 down after 71 minutes with goals from Ellis Simms and Callum O'Hare and a Haji Wright penalty deep into stoppage time.
Second-tier Coventry were the better side in extra time and thought they had pulled off a remarkable win against the 12-time champions when Victor Torp scored from close range 20 seconds from the end of extra time.
The joy of the 36,000 travelling Coventry fans turned to tears though as Torp's goal was disallowed by VAR with Wright adjudged to have been centimetres offside before playing the ball across the face of goal.
United then won 4-2 on penalties to deny Coventry a second FA Cup final, having beaten Tottenham Hotspur in 1987, and prevent them from becoming the first team from outside the Premier League to reach the final since Cardiff City in 2008.
Despite the outcome it felt like a moral victory for Coventry who were magnificent after a poor first hour.
"Bitterly disappointed, but I'm proud as well," said Robins whose goal in an FA Cup third round for United against Nottingham Forest in 1990 is said to have saved then manager Alex Ferguson's job and paved the way for a glittering era.
"Someone said to me I don't think any team have been as close to a final without getting there.
"We deserved to go through, we were 20 seconds away and a toenail offside, it's ridiculous."
Robins said Simms' goal gave his side belief after struggling to make any impact before that.
"We thought it might be a consolation and then the second loops in and it gives us a lifeline. Your tails are up then. Haji sticks a penalty away and all to play for," he said.
"They hit the bar, we hit the bar. Another goal with 20 seconds to go. It took an age. The fourth official said there was nothing wrong with it (initially). A toenail offside."
