Dawes, 60, took temporary charge of the Sky Bet League Two club for the third time after Micky Mellon was sacked in March.
Rovers said in a statement: “The club has considered several highly qualified candidates, however Ian’s coaching abilities, knowledge of the playing squad and the staff as well as the club’s aims to balance player development with winning games ultimately made him the standout choice.”
Andy Parkinson will continue in his role as first-team coach, while the club continue their search for a technical director.
Dawes said: “It’s an honour to be appointed manager of Tranmere. Having already had three interim spells, I’m now ready to take control of the team and implement my own style and ideas.
“It’s an opportunity I’m relishing and I want to reassure the fans that I personally hate failure and would not have taken the job unless I was confident of bringing success to the club.”
Tranmere have climbed up three places to 11th in the table since Mellon’s departure, with Dawes winning two and drawing four of his eight games in charge.
Dawes will be targeting a top-10 finish before Monday’s final game of the season at home against automatic promotion hopefuls Northampton.
With their initial replay called off due to a waterlogged pitch, Tranmere - who came back from 3-0 down to draw 3-3 at Vicarage Road - and Watford met at Prenton Park on Thursday - just two days after the Hornets' Premier League defeat to Aston Villa.
Nigel Pearson subsequently made wholesale changes and, though Kaylen Hinds' 68th-minute effort cancelled out Emmanuel Monthe's first-half opener, Watford succumbed in extra-time.
Substitute Paul Mullin, who scored a late spot-kick in the first meeting between the sides, was Tranmere's hero - heading in the winner to set up a mouth-watering tie with 12-time winners United.
United arrived at Prenton Park having lost back-to-back Premier League games against Liverpool and Burnley and the club's plight prompted some in the away end to reprise the disgruntled atmosphere that filled Old Trafford after defeat to the Clarets four days earlier.
Goals from Harry Maguire, Diogo Dalot, Jesse Lingard, Phil Jones, Anthony Martial and Mason Greenwood lightened the mood among the United faithful as Solskjaer's men breezed past League One opposition, and the manager praised his side's approach after they avoided a potential banana skin.
"If you go into the trap of playing long balls and making this game a fight, we're not used to that," Solskjaer told BT Sport. "They would have been better suited than us. We just asked our players to play and it went well.
"I think the players played football the way it should be played. It was an exceptional performance at times. It was a very difficult pitch.
"It wasn't just the Tranmere people who wanted an upset, there were a lot of others. We've just got to stay together and keep working.
"Every game you win is a good one for your confidence. Every time you score a goal and perform well, it's good.
"The attitude was spot on. That's all you can ask for. When the attitude is right, you can't fault them."
Maguire, who opened the scoring for United with a powerful 20-yard drive, and then provided the assist for Lingard to score their third in a display full of guile and attacking intent, said his first goal for the club was long overdue.
"It's been a long time coming," said Maguire. "I've had a lot of chances this year without finding the back of the net. It’s nice to help the team get into the next round.
"It was really important to get the win, coming off the back of a bad performance and result against Burnley.
"We set our stall out to put in a positive performance for the fans. We're in the next round, which is what we came to do."
Her Majesty passed away at her home in Balmoral on Thursday at the age of 96.
Norwich City were scheduled to visit Burnley in the Championship on Friday, while Tranmere Rovers and Stockport Country were set to meet in League Two.
Both games have now been called off, with the EFL confirming a decision will be made on the rest of the weekend's games on Friday.
A statement on the EFL website read: "As a mark of respect, following the passing of Her Majesty, Queen Elizabeth II earlier today, the EFL has confirmed that its fixtures scheduled for tomorrow evening (Friday 9 September) – Burnley v Norwich City and Tranmere Rovers v Stockport County - have been postponed.
"A determination regarding the remainder of this weekend's scheduled fixtures will be made following a review of the official mourning guidance, in addition to further consultation with DCMS [Department for Digital, Culture, Media and Sport] and other sports on Friday morning."
