The 36-year-old led part-time Braintree to the play-offs, where they lost to Grimsby, in his first season in charge.
Cowley, who had combined his Iron job with work as a PE teacher, has signed a two-year deal and will be assisted by his brother Nicky at Sincil Bank.
Lincoln were without a boss after Chris Moyses left at the end of the season.
Former Concord Rangers boss Cowley previously said he wanted to work in full-time management and is set to take over at a club that was in the Football League as recently as 2011.
Braintree chairman Lee Harding told BBC Essex: "Being fair to Danny and Nick, if they think this is the right opportunity for them going forward, we understand and support it.
"This is not about what they're doing as their hobby, this will be their full-time job and if they decide the package they're being offered is worthwhile giving up what they do in their day job then it's a big step for them and I hope they get it right.
"With the news that broke earlier in the week, we've already had people calling asking us to think of them which is nice. We will advertise the post and I would imagine we'll have a lot of applications."
"It's no surprise the Cowley brothers have caught the attention of other clubs after three promotions with Concord and a stunning third-place finish with Braintree on one of the National League's smallest budgets.
"The slight surprise is it being Lincoln who've managed to snare them. Other clubs were rumoured to be interested with Tranmere and Notts County in the frame.
"But it's Lincoln who showed the courage of their convictions and put money on the table. It's a risk for the brothers with Lincoln's budget by no means the biggest and they're now at a club with higher expectations. But, knowing the amount of detail they go into and hard work they put in, I would expect them to be a success."
Danny Cowley has been appointed as Lincoln City's new manager after resigning as boss of fellow National League club Braintree Town.