After Voldemort's fall in October 1981, his entire army has collapsed. Almost all of his Death Eaters have surrendered to the Ministry claiming to be under Imperius, thus escaping punishment. The few who persisted were imprisoned in Azkaban. While the fate of lower-ranking, unmarked servants is not described explicitly, there are hints to them being partially imprisoned and partially scattered: Sirius claims about a bunch of Azkaban inmates hating Karkaroff for rating them out and Fenrir seemingly remains at large, but doesn't continue the fight.
What is remarkable about this situation is the Ministry's leniency towards Voldemort's middle management. During the war the Ministry authorized torture, enslavement, indefinite imprisonment and murder of anyone suspected of following Voldemort. The Ministry didn't care about someone being under Imperius, or being rich, or being from a prominent family. Nor was there some kind of leadership or policy change after the war to explain the Ministry's sudden leniency: the same Minister remained in power for a decade till 1991 and despite war's end there was no attempt to investigate cases of people thrown into Azkaban without trial.
Sure, one might try to explain it all away by claiming corruption, but if gold was all it took to claim innocence-via-Imperius, how come Rookwood and Dolohov were left to rot in Azkaban while Macnair did not? Was Macnair, an executioner, that much richer than a Department of Mysteries empoyee Rookwood? And why would the Ministry care to take Death Eaters' gold in exchange for their freedom if it could just take all of their gold to the last Knut and leave them to rot for treason and rebellion?
Barring a sudden outbreak of post-war naivete, how come the Ministry just let most of Voldemort's lieutenants go?
Consideration 1: the state of the Ministry
While 1981 Ministry was never quite as pathetic as to allow itself to be couped by Voldemort, every description of the war provided to us paints a rather grim picture of a decade-long war where the Ministry was slowly losing ground. Despite emergency measures championed by Barty Crouch Sr, Voldemort was gaining strength, his army was growing via recruitment/bewitchment/intimidation, while his enemies were fading - opponents intimidated, families wiped out one-by-one with the Dark mark left in the sky to remind everyone what happened to people defying the Dark Lord.
It is most probable that after a decade of fighting the war - and almost losing - the Ministry was in no shape to continue the fight against Voldemort's minions, even if Voldemort himself was removed from the board.
Consideration 2: the state of Voldemort
In a great contrast to 1995, 1981 Ministry - and indeed society at large - seems to believe that Voldemort is not finished and that he may return. When Crouch Sr accuses his son & co of conspiring to bring Voldemort back, it's treated as a valid prospect, and Voldemort himself mentions how the Aurors were searching for him between wars.
With the prospect of Voldemort's immediate return in mind, it would be perfectly logical to break up his army by granting clemency to his followers, and focus on looking for the Dark Lord and preventing his return to power.
Consideration 3: the state of Death Eaters
During the first war, Death Eaters used hit-and-run tactics to fight against the Ministry: Dark wizards apparate in, torture/enslave/murder using superior numbers, leave the Dark mark for extra terror and disappear again. The Ministry was obviously at a disadvantage since it couldn't be everywhere at once, thus allowing Death Eaters to eliminate their enemies piecemeal.
What was to stop Death Eaters from continuing the same thing without Voldemort?
Absolutely nothing. Having Voldemort was certainly a massive advanatage, but it's not like Death Eaters couldn't repeat the same strategy without him. And that's something the Ministry was bound to tak into account while considering how to proceed after Voldemort's fall.
Consideration 4: Death Eater morale
Voldemort's army had very bad morale. It was held together by fear. Most people - even objectively bad people who liked torturing, killing and enslaving - didn't want to be there by 1981. Nobody is happy to see Voldemort back in 1995, and that was a reuinion of marked Death Eaters, the chosen servants. Sure, there were Lestranges and Crouch Jr, but most Death Eaters are very happy never to see Voldemort again.
In short, most Death Eaters were willing (and eventually did) abandon Voldemort once he fell from grace and wasn't there to threaten them anymore. All the Ministry had to do was to make sure that fear of being murdered by Voldemort for desertion was not replaced by fear of being fed to the Dementors.
Conclusion: it was a deal
The Ministry didn't just develop a sudden case of mercy 1981. The story about Death Eaters being pardoned because they were placed under Imperius curse - it was just that, a story. A hastily invented, very stupid cover story, a little fig leaf to hide the deal made between the Ministry and Voldemort's middle management.
People like Malfoy, MacNair, Goyle and the rest laid down arms in exchange for pardons. Given how all hardliners were quickly caught and placed into Azkaban, it's very probable that turncoat majority helped the Ministry to catch their hardliner compatriots. Maybe not directly, but by providing information where and when hardliners planned to go next. Deprived of leadership, the rest of Voldemort's army collapsed overnight.
The Ministry could dedicate its meager remaining resources to looking for Voldemort (which again, they did according to the man himself) for the next few years until complacency finally took hold and most people persuaded themselves that Voldemort was gone forever.
And Karkaroff - a foreigner with no connections or family in Britain - was conveniently made the public scapegoat, allegedly providing all that juicy information which led to arrests and imprisonments. Naturally Karkaroff had no desire to test fate by waiting around for some remaining fanatic to punish him for his betrayal, so he buggered off to the continent, eventually becoming a headmaster in a foreign school.
Clemency for Death Eaters was a sound (if cynical) short-term decision prioritizing a larger threat... but only on the condition that the Ministry would remain on its toes and prevent Voldemort from returning. And the Ministry grew complacent instad, allowed Voldemort to resurrect himself, at which point the decision became a giant liability since Voldemort had most of his key followers ready to go at the moment's notice.