The Magpies were looking comfortable at Meadow Lane after goals in each half from Garry Thompson and Liam Noble.
But a 90th-minute penalty from Nathan Tyson, awarded after a foul on the Doncaster striker by Haydn Hollis, gave the visitors hope but the hosts held on.
With Crawley also falling to a late defeat, it means County move out of the relegation places on goal difference and manager Moniz conceded his side’s fast start had been vital.
“We scored early and you need that because if you concede you are chasing, but on this occasion we took the initiative,” he said.
“After 30 minutes we ran back too much and gave too much space to their midfield.
“(Dean) Furman was free, (Harry) Forrester went to the right and our organisation went.
“We must organise better. We must play like we are in the streets. We say ‘I’ll take him, you take him’.
“They showed they could play good football but after we went 2-0 up we didn’t do what we had spoken about.
“They scored a penalty in the last minute and then you have a heart-attack.
“I can’t have any influence over the penalty, but we showed a lack of discipline beforehand.
“But the fans got what they deserved. We gave them something and we must win their respect. They responded and it was pretty unique.”
Rovers, safely in mid-table, were slow out of the blocks and were punished when Thompson converted Graham Burke’s cross.
The visitors rallied but went further behind when Noble finished from close range in the 60th minute.
While his side pulled a goal back late on, visiting manager Paul Dickov admitted they had not been at the races.
“There was one team fighting and scrapping for their lives and the final outcome was probably the right one,” Dickov said.
“They just had more energy and they had more to play for because they are trying to stay in the league.
“We started the game with a diamond formation because we thought that would cause them problems.
“But when you play a diamond you have to play through the thirds and we just kept lumping it to Nathan Tyson.
“We changed it around and put James Coppinger in the middle and I thought he did really well in there.
“But the goals we conceded were very poor and the aim for us now is to go into the last game of the season, which is at home, and finish the campaign off.
“Once we’ve done that then we can start the work that we need to do.”