Mr McConnell resigned after losing the elections to the SNP in May.
Ms Alexander eventually emerged as the only candidate to replace him but had already raised funds to run a leadership campaign.
She previously said she had been told wrongly by clerks to the standards committee it was unnecessary to declare the donations.
She later updated her register with details of 10 donors, who each gave about £1,000 to her campaign.
This was after the Scottish parliamentary standards commissioner, Dr Jim Dyer, decided the donations should be treated as gifts.
The Holyrood standards committee decided on Thursday to recommend that she be suspended from parliament for one day.
Ms Alexander called that decision "partisan" and said she was certain MSPs would overturn it when they returned after the summer recess.
Speaking at Scottish Labour's headquarters in Glasgow, Ms Alexander said: "My pursuers have sought the prize of political victory with little thought to the standing of the parliament.
Wendy Alexander talks of 'distractions' leading to her resignation
"Some may feel they have achieved a political victory but wiser heads will surely question 'at what price?'"
Ms Alexander added: "It is clear that vexatious complaints will continue to dominate the headlines as long as I remain Labour's Scottish Parliamentary Leader.
"I cannot ask Labour supporters in Scotland for further forbearance."
She later told BBC Scotland that she thought the parliamentary process had been "cynically abused" by political opponents who had used "investigation as political tactic".
Ms Alexander admitted she had made mistakes but said there had also been "politically-motivated" complaints".
Scottish Labour's deputy leader Cathy Jamieson has taken charge ahead of a leadership contest.
The prime minister said: "I want to thank Wendy Alexander, not just for her work in rebuilding the Scottish Labour Party since the last elections but for her commitment to devolution and her part in establishing the Scottish Parliament.
"Her dedication to social justice can never be doubted and her contribution - which has been outstanding over the years - will be greatly missed."
Decay from within is characteristic of the decline of the New Labour project
Scottish deputy first minister Nicola Sturgeon acknowledged it was a difficult decision for Ms Alexander.
The Scottish National Party deputy leader said: "While Wendy Alexander has been author of own misfortune, there can be no doubt that the information on her illegal campaign donation could only have come from within the inner circles of the Labour Party.
"Decay from within is characteristic of the decline of the New Labour project, and Wendy Alexander's resignation is a symptom of this wider malaise."
Scottish Conservative leader Annabel Goldie said: "This resignation is a further body blow for Gordon Brown and Labour, as they lurch from one crisis to another.
"Only the Scottish Conservatives offer a credible opposition to the SNP in Scotland."
Scottish Lib Dem leader Nicol Stephen said: "It is sad that Wendy Alexander has been forced to resign in these damaging circumstances.
"She has always been respected for her intellect and her commitment to a stronger Scottish Parliament.
"But this donations issue has dogged her leadership from the very start."
The standards committee ruling was the latest development in the saga of donations to Ms Alexander's leadership campaign.
In a separate development, she earlier expressed "deep regret" that her team accepted an illegal £950 donation from Jersey-based businessman Paul Green.
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