Winger Mackay-Steven, 24, had signed a pre-contract to join the Parkhead side in the summer but after the two clubs agreed a fee late on Monday night he penned a four-and-a-half year deal.
Midfielder Armstrong, 22, signed a three-and-a-half-year contract, with the combined fee of £2million plus add-ons reported.
The departure of two of United’s best players left many fans in despair, two days after they celebrated the 2-1 League Cup semi-final win over Aberdeen at Hampden Park, even though McNamara’s squad has been replenished ahead of the March 15 final against the Hoops.
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Robbie Muirhead and Paul Dixon were recruited on transfer deadline day, joining earlier signings Henri Anier, Ryan McGowan and Simon Murray, with an extended deal confirmed in December for Nigerian Under-23 international Ola Adeyemo.
The United board issued a statement on the club’s official website to enable fans to have “a complete understanding of the club’s current position” while highlighting McNamara’s backing.
The statement read: “Gary Mackay-Steven and Stuart Armstrong left for Celtic FC and while we understand the initial negative response in some quarters to their departure from Dundee United, it is very important to keep this in context.
“Firstly and most importantly, it is the board’s duty to balance the short, medium and long term interests of the club at all times.
“While being an emotional wrench for the club and the players, the transfer of Gary and Stuart was reluctantly, albeit undoubtedly, the right thing to do in the circumstances.
“Those circumstances, for absolute clarity, are that both Gary and Stuart made it clear to management that they wanted to leave Dundee United to further their careers elsewhere.
“Gary had signed a pre-contract agreement with Celtic.
“The board had no desire to sell the players. However, when it was obvious that they both viewed their futures elsewhere it is the board’s responsibility to do the right thing for the club.
“That then meant ensuring that the club realised optimum financial value for the players - one of whom had only 17 weeks left on his contract.
“It should also be appreciated that, in reality, players decide where they will play and not their current clubs. Dundee United has no control over their destination club.
“On this occasion the lure of European football and increased personal financial reward and security are major draws to young players. Both players ultimately go with our best wishes.
“Such deals are negotiated at all times in the longer term interests of the club.
“This is also a very important point as the club remains focused off-field on being debt free.
“The transfer fees received will have the incidental effect of assisting the club to become completely debt free ahead of schedule.
“On-field, we are still challenging on three fronts and have no intention of being diverted from our short-term aims for this season.
“We can say that safe in the knowledge that the manager himself is happy with the result of this transfer window and fully supported the board’s actions.”
The statement continued: “We have some of the best players in the country and we have the best management team for attracting the best young talent.
“We trust in the manager, his coaching staff and the players as an entire squad.
“As a board, we are optimistic about the club’s future as we know the structure of the club and the personnel that are committed to the club, both on and off field, are acting in the best interests of Dundee United at all times.”
The statement also denied reports that the club was, “interested or involved in talks to buy a football club in Australia”.