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Harry Sawyer scored the only goal of the game on 69 minutes, and although Diamonds strived for an equaliser, they were hampered by the 84th minute red card for Sidik Atcha and could find no way back.
The result United leapfrog Diamonds in the Pitching-In Northern Premier League Midlands Division, with Elliot Sandy's side slipping to 13th - six points off the top seven.
The Diamonds boss had been hoping his team would get their league campaign back on track after the end of the Emirates FA Cup run at Burton Albion on Saturday, but he was left bitterly disappointed.
"At Coventry we looked off it from the beginning," Sandy told @AFCRD.
"We gradually grew into the game but we just looked really vulnerable defensively, every time they turned it in behind.
"They have got really good, quick lads who are always dangerous and made it really difficult for us.
"But we were just sloppy all over, and it is really disappointing.
"Teams that do well in leagues, come to places like this, on nights like this, and show up, and we didn't."
The match was Diamonds' sixth away game on the spin in all competitions, but Sandy is not using that as any sort of excuse.
"We were happy to go in at 0-0 at half-time, we changed things around, and I thought we were more in control in the second half," said the Diamonds boss.
"We had a lot more of the ball and they weren't hurting us, but the thing was we weren't really hurting them.
"They didn't want to concede, they kept a great shape, kept the ball in front of them, and we just lacked ideas.
"Our lads that have been so good for us at the top end of the pitch, I thought they were poor, and defensively we looked a bit all over the place as well.
"In midfield we didn't use the ball well, so we were disjointed in all three areas, and then we concede a really poor goal.
"We then come to life and are trying to press, but I said to them we need to be like that from minute one, we need to get after teams and show that desire that we have done, particularly at home.
"This was out sixth away game in a row which obviously is a lot in the lads legs and in terms of travelling, but it is no excuse.
"We have got a really good squad and they need to be better."
Diamonds are back on the road on Saturday, for the seventh game in a row, when they go to St Ives Town.