The bottom side looked like earning at least a temporary reprieve as they opened up a 2-0 lead with goals from Habib Beye and Fabien Robert inside the opening five minutes.
But all began to go wrong as Beye conceded the first of two penalties early in the second half to earn a red card.
Greg Halford tucked away both spot-kicks and, although James Coppinger put Doncaster back ahead, Pompey substitutes Dave Kitson and Marko Futacs sealed their fate in the closing minutes. Portsmouth remain seven points off safety with just three games left.
Appleton said: “Our chances of staying up are not great. Even if we win the three games, all Bristol City have to do is look after their own ship. I think if it was a Grand National we’d be a 500-1 shot.”
The other side in the bottom three, Coventry, could only manage a 1-1 draw at Burnley but their manager, Andy Thorn, retained hope of survival with his side four points off safety.
Charlie Austin gave the Clarets a 20th-minute lead at Turf Moor with the Sky Blues, who are four points from the crucial 21st place, replying through Clive Platt.
Thorn said: “[To] still be in there with two home games to come is great.’’
At the other end of the table West Ham put themselves back in the hunt for automatic promotion with a 6-0 hammering of play-off hopefuls Brighton, putting them within three points of second-placed Southampton.
Ricardo Vaz Te set up victory with two goals inside the opening eight minutes and completed his hat-trick just after the hour.
Kevin Nolan was also on target as the Hammers established their early stranglehold while Carlton Cole struck and Gary Dicker turned into his own net in the second half.
Boss Sam Allardyce said: “We needed three points and to win in such style was thrilling.’’
Sixth-placed Cardiff gained ground in the play-off race as Birmingham and Blackpool, in fourth and fifth respectively, were held to draws by Bristol City and Nottingham Forest.
Ireland midfielder Liam Lawrence fired Cardiff’s winner after 69 minutes.
* Former Cardiff manager Eddie May died on Saturday at the age of 68. May led the Bluebirds to the Division Three title in 1993.