DUP leader Edwin Poots formally nominated Givan, the Lagan Valley MLA, at a special sitting of the Stormont assembly on Thursday.
Sinn Féin’s Conor Murphy nominated his party colleague O’Neill to take up the role of deputy first minister.
Givan thanked his party leader for having “confidence in me”.
The process went ahead despite a morning of uncertainty and unease from senior DUP figures who questioned their party leader’s decision to proceed.
It is understood a significant majority of DUP MLAs and MPs voted against Poots’s decision to nominate a Stormont first minister.
The vote was taken after a heated internal party meeting at Parliament Buildings ahead of the nomination process in the assembly.
PA understands that Poots and Givan had left the room before the vote took place.
One senior party source at the meeting described the atmosphere to PA.
“Dreadful. Utterly dreadful. Never experienced the like of it,” said the source.
On Thursday morning, party MPs and peers sent an urgent email to Poots urging him to hold off nominating Givan until he explained his decision to reconstitute the powersharing administration after Sinn Féin secured a key concession on Irish language laws.
The email, a copy of which has been seen by the PA news agency, is signed by defeated leadership candidate Sir Jeffrey Donaldson, party chairman Lord Morrow, senior MPs Sammy Wilson, Gregory Campbell and Gavin Robinson, former deputy leader Lord Dodds and a number of other senior members.
In total seven of the DUP’s eight MPs signed the email, with Ian Paisley being the exception. The party’s five peers also signed it.
Many of those who signed the email would have supported Donaldson in his leadership bid, though some, like MP Paul Girvan, supported Poots’s candidacy.
The email read: “We note the announcement made by the secretary of state in the early hours of this morning that both you and the Sinn Féin leadership have agreed to nominate a first minister and deputy first minister on the basis that Westminster will legislate on the Irish language and other matters if the assembly fails to do so by October. We are also in receipt of your email this morning regarding this agreement.
“We are very concerned about this development and therefore, are urgently requesting that you meet with us as DUP members of parliament and peers to explain the basis of your agreement with the secretary of state and Sinn Féin before any further steps are taken in this process, including the nomination of a first minister. Assuming you will have prior consultation with your assembly group, we would be happy to join this meeting.”