Specifically, Johnson was questioned about whether the DHS helped the DNC after learning about possible Russian hacking.
"The FBI and the DNC had been in contact with each other months before about the intrusion," said Johnson. "The DNC did not feel it needed DHS's assistance at that time."
"Why did Democratic National Committee turn down the DHS offer to protect against hacks (long prior to election). It's all a big Dem HOAX!" he wrote.
AT Thursday's press briefing, principal deputy press secretary Sarah Huckabee Sanders attempted to clarify Trump's use of the word "hoax."
"I believe that the reference to the hoax is about the fact that they're trying to delegitimize his win," she said of the Democrats.
Former DNC Chairwoman Debbie Wasserman Schultz contradicted Johnson's account, issuing a statement Wednesday that read, "At no point during my tenure at the DNC did anyone from the FBI or any other government agency contact or communicate with me about Russian intrusion on the DNC network."
"It is astounding to me that the chair of an organization like the DNC was never contacted by the FBI or any other agency concerned about these intrusions," she added.
In a third tweet Thursday morning, Trump asked, "By the way, if Russia was working so hard on the 2016 Election, it all took place during the Obama Admin. Why didn't they stop them?"
At a forum hosted by the House Homeland Security Committee on Thursday, Johnson responded to Trump's tweets but did not say whether he felt the president had twisted his words, adding that he would "leave that to the journalists."
He advised people to focus on the current administration's actions around Russian interference rather than past efforts by the DNC or Obama administration, saying, "The larger question that we need to address is, now that we know what happened, what are we going to do about it ... to stop a foreign superpower from interfering in our democracy?"
He added that he believes that America remains "exposed."
Johnson said the country "has not done much to harden" its cyberdefenses and stressed that a more robust defense would make hacking U.S. systems "cost prohibitive" for other countries.
ABC News' Alexander Mallin contributed to this report.