A tight first half came to life just a minute after the restart when Daryl Murphy headed in his 13th goal of the season to put the hosts ahead.
Chris Brown equalised from the penalty spot with seven minutes remaining, but there was still time for Mick McCarthy's men to claim all three points as Luke Chambers fired in just three minutes later - following an incident which had Dickov screaming that his right-back, Gabriel Tamas, had been fouled by Luke Hyam in the build-up.
Dickov said: "I feel really hard done by. It was a blatant free-kick on Tamas - he was lucky not to have his leg broken.
"Tamas has six studs down his shin. It is the inconsistency that is so frustrating. I am going to see the referee to talk to him. I can't tell you what I am going to say - it would be unprintable.
"Ipswich are a fantastic club with a fantastic manager. I don't want to see anyone get sent off, but it wasn't just a foul - it was a red card."
While Doncaster continue to fight for their Sky Bet Championship survival, Ipswich are now only out of the play-off spots on goal difference.
After a tense first half, with Tamas and Town centre-back Tommy Smith both spurning good chances, Ipswich looked to be heading to a comfortable three points - Murphy heading home just one minute and seven seconds after the restart.
Doncaster clawed their way back when Tyrone Mings' clumsy foul on Mark Duffy allowed Brown to calmly find the net from 12 yards.
But Chambers restored Ipswich's lead before a frantic finale which saw Duffy denied by Dean Gerken and Frank Nouble miss a glorious chance to make the game safe.
McCarthy admitted that the Doncaster defender had been fouled in the build-up to the winner, but he only realised after seeing a replay of the contentious incident.
McCarthy, who also described Doncaster's penalty as a stonewall, said: "It was a must-win. There are big games coming up and it is exciting to be part of it. We have a chance of getting promoted to the Premier League.
"If you can get in the play-offs, one team has to win. It might be a side who gets sixth and has momentum - like us, or Bournemouth or Brighton. We would all provide difficult opposition and the play-offs are such a lottery.
"People recognise the spirit in the team but we also have good players. The defence have chipped in with goals and Luke Chambers has been excellent, having played out of position all season."