Celtic had offered United £250,000 to take winger Mackay-Steven, 24, now rather than wait until the summer, a move that had already been agreed.
And the Glasgow club agreed a deal for both players of £2m plus add-ons.
"I'm delighted to be here and to have signed on at such a massive club," Armstrong, 22, told BBC Scotland.
Mackay-Steven added: "I'm absolutely delighted to be here and to get everything done and dusted today. It's been a crazy few hours but I'm over the moon."
Celtic's increased £1.6m offer for Armstrong had been rejected on Sunday before further talks took place on Monday.
The Scottish champions had a £1.5m offer for the Scotland Under-21 international rejected last week and Burnley also had a bid turned down.
The £250,000 offer for Scotland cap Mackay-Steven was £100,000 more than they had previously offered.
Armstrong, who has signed until 2018, and Mackay-Steven, whose deal runs until 2019, will be ineligible for Celtic's Scottish Cup meeting with Dundee on Saturday and next month's League Cup final meeting with their former club.
United defeated Aberdeen 2-1 at Hampden Park in Saturday's semi-final, with Celtic recording a 2-0 victory over Rangers the following day.
And United will ask for at least 17,500 tickets for the 15 March showpiece.
The Tannadice board will also request that tickets outside of the corporate and main stand be priced at a maximum of £25 for adults and £10 for concessions.
One director told BBC Scotland: "We want to ensure as big a crowd as possible for the game, which we are sure the sponsors would want too.
"The game is being played in winter and, to make sure the game attracts as big a crowd as a possible, we will ask for tickets to be priced at a reasonable level.
"That is very important for the fans of both clubs when money is tight."
It is believed that United will also ask for the Toryglen end of the stadium to be reserved for their fans at a planning meeting on Tuesday with the Scottish Professional Football League about the final.
However, BBC Scotland has learned that the SPFL will stick with tradition and allocate that end of the ground to Celtic fans.

Neil Fears, from Belper in Derbyshire, was reported missing by his fellow divers on Tuesday after he failed to surface.
The 51-year-old's body was recovered near the wreck of the SS Stanfield, near La Manga, about three hours later.
It is not yet known why the experienced diver failed to surface.
His daughter Cerys said: "He was always there if I needed him for whatever situation, he'd do anything to look after his family."
Mr Fears was a director of IT firm Davidson Richards in Derby.
"He was very hardworking and could be very determined," his daughter added.
"My boyfriend said he was strong and ambitious, but he was also very kind and caring, generous and very clever."
Mr Fears had been diving with three others and was paired up with one of them.
A spokesperson for the Guardia Civil in Murcia said the paired diver had "turned to his partner and realised he wasn't there".
The divers had been using equipment which regulated oxygen supply and pressure, depending on the user's depth under water.
This meant they had to return to the surface regularly, but Mr Fears failed to do so as planned.

Avon and Somerset Police were called to the junction of Creswicke Road and Airport Road in Knowle West at about 21:05 GMT on Tuesday after reports a man had been run over.
Officers said the crash involved a car and a pedestrian, aged in his 20s, who died from his injuries.
A 30-year-old man has been arrested on suspicion of causing death by dangerous driving and is police custody.
The road has reopened but a nearby petrol station on Hengrove Way remains sealed off while investigations continue.
A spokesman said: "A full investigation has been launched and one of our priorities is to ensure the victim's next of kin are given all the support they need."
Midfielder Stuart Armstrong and winger Gary Mackay-Steven have joined Celtic from Dundee United after the clubs agreed deals on transfer deadline day.
[NEXT_CONCEPT]
The daughter of a British diver who died while exploring a World War One shipwreck off the coast of Spain has paid tribute to her "caring" father.
[NEXT_CONCEPT]
A pedestrian has been run over and killed in Bristol.