The Republic of Ireland international signed a three-and-a-half-year deal at Turf Moor for an undisclosed fee.
But the Clarets confirmed the sum eclipsed the £10.5million they paid Derby for Jeff Hendrick last summer.
Brady spent 18 months at Norwich, having joined from Hull for £7million, and was part of the team relegated from the Premier League last season.
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He was the second new arrival at Burnley on Tuesday with Sean Dyche also signing Ashley Westwood from Aston Villa.
Southampton finally got their hands on Italy forward Manolo Gabbiadini and Leicester signed Mali international Molla Wague on loan from Granada.
Burnley were one of the busier sides in the Premier League in an otherwise quiet deadline day although Odion Ighalo completed one of the other biggest deals after leaving Watford for China.
The striker joined Chinese Super League club Changchun Yatai for an undisclosed fee, although it was reported to be around £20million.
Emmanuel Adebayor joined Turkish Super Lig team Istanbul Basaksehir almost seven months after leaving Crystal Palace as the January transfer window deadline day began slowly.
The striker, 32, passed a medical and signed a reported 18-month contract.
Adebayor, formerly of Arsenal, Manchester City and Tottenham, was released by Palace last June at the end of his six months there.
Second-placed Istanbul Basaksehir are presently second and represent Adebayor’s first permanent non-English employers since he arrived at Arsenal from Monaco in 2006.
“Emmanuel Adebayor passed the health screening at Medipol Mega University Hospital,” the Turkish club wrote on Instagram.
“It has been completed to include health checks, chest X-ray, cardiac tests, and ultrasound.”
Premier League strugglers Hull have also announced the loan signing of Italy international defender Andrea Ranocchia from Inter Milan.
Ranocchia, 28, who could make his debut for Hull at Manchester United on Wednesday night, is their sixth signing of the transfer window.
He joins fellow loan recruits Oumar NIasse, Evandro, Omar Elabdellaoui and Lazar Markovic. Markus Henriksen had also arrived for an undisclosed fee from AZ Alkmaar.
Deadline day was otherwise proving uneventful during its opening hours.
The January window closes at 11pm in England, and few moves of significance had been agreed by late morning on Tuesday.
Fourteen of the division’s 20 teams are involved in Premier League fixtures on Tuesday night which perhaps contributed to the lack of activity.
Crystal Palace manager Sam Allardyce on Monday spoke of his desire to strengthen his squad, but insisted he would be focusing on their trip to Bournemouth instead of transfers.
Championship Blackburn have re-signed Marvin Emnes, 28, on loan from Swansea until the end of the season.
Jordi Gomez, meanwhile, has left their rivals Wigan to join Spain’s Rayo Vallecano, signing a contract until the end of the season.
One player hoping to secure a deadline day move is Leicester’s Leonardo Ulloa, who said he felt “betrayed” by Leicester boss Claudio Ranieri on Monday and vowed not to play for the club again.