Humza Yousaf has denied the SNP was "in cahoots" with Police Scotland over the timing of the arrest of its former chief executive.
Peter Murrell, the husband of Nicola Sturgeon, was quizzed by detectives yesterday before being released without charge last night.
At the same time, dozens of officers searched the SNP headquarters in Edinburgh as well as the house belonging to Sturgeon and Murrell in Glasgow.
The raids were carried out during the Scottish Parliament Easter recess and just a week after Yousaf was announced as the new leader of the SNP.
They were conducted as part of a long-running police investigation into SNP finances.
Alex Neil, a former Scottish Government minister, said today he thought Yousaf would have "found it much more difficult to win" the leadership race "if all of that had happened prior to the ballot opening".
Speaking to journalists at Bute House, the Record asked if speculation regarding the timing of the police investigation had called into question the leadership result.
The First Minister said: "To me that sounds like a conspiracy theory that somehow we were in cahoots with Police Scotland around the timing. The timing of any investigation is absolutely for Police Scotland. It's not determined by anybody else."
But Yousaf admitted that SNP members would be feeling "bruised" following recent events.
He continued: "There was a six week battle that took place, a battle of ideas, in terms of policy positions, and the members put their trust in me.
"There is no getting away from the fact our members will feel bruised. Not just from a challenging election contest, but the events of the last 24 hours as well. And my job is to try to give them as much confidence as I can about changes we are going to make internally."
Yousaf also admitted it was "very, very clear that the governance of the party is not as it should be".
He described a row over the publication of SNP membership numbers as a "debacle".
"I think whatever else transpires in this case, it's very, very clear that the governance of the party is not as it should be, is not at it's best standard. Frankly, the debacle over membership numbers is just one example of that.
"I'm not going to make this personal about Peter or Nicola. I have respect for both.
"But clearly there is the opportunity, with a new chief executive, there is an opportunity to do things differently."
Alex Neil, a former SNP MSP, had earlier told BBC Radio 4 that the on-going police investigation into party finances was "likely" a factor in Sturgeon's resignation in February.
He said: “Given that a large number of people had been interviewed, including staff at SNP headquarters, it’s hard to believe Nicola wasn’t aware that the enquiry was coming to a head to some extent.
"If that was the case, then you could understand why she would not want to be First Minister when this happened.
"I can't say one way or the other... but I think most people would conclude that it might well have been, was probably likely to have been a factor in her decision to resign at the time that she did."
The former government minister added: "Humza stood as the continuity candidate. Had this all happened, including the row about the party membership, the refusal to release the details of the party membership.
"If all of that had happened prior to the ballot opening, I’m absolutely sure that the continuity candidate would have found it much more difficult to win.”
Jackie Baillie, Scottish Labour depute leader, said: "After staying silent for months while people across the political spectrum raised the alarm over the culture in the SNP, Humza Yousaf has finally come clean and admitted that there are significant issues to deal with.
"It’s plain for all to see that Humza Yousaf was happy to turn a blind eye to the debacles in the SNP for as long as he could.
"You simply cannot trust Humza Yousaf to deal with the culture of cover-up at the heart of the SNP.
"Scotland deserves better than the scandal-struck SNP and it’s out of touch and out of depth leader.”
