Wrexham, who had trouble finding a squad fit enough to play, worried the Tigers towards the end but could not do enough to secure a vital point in their battle to avoid relegation.
City had started brightly and Wrexham had a scare in the 10th minute when Alex Smith mis-timed a pass back to keeper Ben Foster who did well to kick the ball away.
Minutes later, Foster held a 20-yard effort from Andy Dawson as Wrexham's defence began to look a little shaky.
Wrexham had a real let-off in the 17th minute when Stuart Green really should have given the Tigers the lead after shooting across goal from just yards out after being played in by Jason Price.
Three minutes later, Chris Armstrong was just as close at the other end, heading inches wide after an excellent ball in from Darren Ferguson.
Allsopp was also guilty of missing a golden chance on the half-hour mark, when, from just six yards out with the ball at his feet and only the keeper to beat, he failed to get the vital touch and a relieved Foster was able to collect the ball.
He redeemed himself five minutes later to give City the lead - but only after a mistake from Foster who spilled a rather tame header from Green's corner. Price was also lurking, but it was City's Australian striker who finally bundled the ball in.
Nick Barmby was booked in the 37th minute after vocally showing his disgust at a two-footed challenge by Dennis Lawrence, who, to the anger of City's fans, was not even shown a card despite the ferociousness of his tackle.
Allsopp saw another effort cleared off the line minutes before half-time after Barmby had played an excellent ball down the line to Price.
Price arguably should have done better with two efforts in the second half, shooting straight at Foster from 18 yards before nodding the ball tamely to the keeper after being given a free header.
But in the 60th minute, Allsopp grabbed his second from 15 yards with a clinical shot after being played in by Elliott.
Substitute Hector Sam threw Wrexham a lifeline in the 75th minute, exploiting some slack defending to fire a low, long-range shot past Boaz Myhill.
How Wrexham were not on level terms in the 80th minute was hard to fathom after a period of chaos in City's box saw what seemed like every Dragons player have a shot on goal.
Virtually all of the last five minutes were played in City's box as Wrexham battled gamely for a point.
Ferguson nearly sealed it with a superb 25-yard strike which curled just over the bar but it was not to be for the Dragons.