The deal is reportedly worth $2million and Rudolph can earn an additional $1.5million in incentives.
Gronkowski retired for the second time in his career last month despite having the chance to play another season with Tom Brady, who announced his retirement on Feb. 1 before changing his mind March 13 and returning to Tampa Bay.
In Rudolph, the Buccaneers land a tight end who had 26 catches for 257 yards and one touchdown last season for the New York Giants.
That was Rudolph’s only year with New York after spending his first 10 seasons with the Minnesota Vikings, who selected him with the 11th pick of the second round (43rd overall) in the 2011 NFL Draft.
Rudolph earned two Pro Bowl selections while playing for Minnesota, including being named the MVP of the game in 2012 after having five catches for 122 yards and a touchdown.
Rudolph has appeared in 156 regular-season games, hauling in 479 passes for 4,745 yards and 49 TDs.
Quarterback Cousins engineered a game-winning drive at the start of overtime in the Wild Card clash, capping it off with a four-yard pass to tight end Kyle Rudolph at the back of the end zone.
The successful throw clinched an upset 26-20 triumph for the Vikings on the road, meaning they move on in the postseason to face the San Francisco 49ers next.
Cousins' career has been blighted by suggestions he struggles in big games, yet Zimmer felt his starter showed he can thrive under pressure, keeping Minnesota's season alive in the process.
"They said he can't win a playoff game. He's only been in two, so he's 50 per cent – that's better than most people," Zimmer told the media.
"I thought he took care of the ball really well today and made good decisions. They had some heat on us, so he had to make some great decisions. He has to go out and prove it again next week, just like we all do.
"Kirk's a good player. The team rallied around him, they're happy for him."
Cousins rejected the opportunity to hit back at the doubters during his news conference, saying: "I'm just thrilled we won a playoff game."
The Vikings had let slip a 10-point lead in the fourth quarter, Will Lutz's field goal with two seconds remaining levelling the contest.
However, once in overtime, Zimmer was determined to make sure his team capitalised on having the ball first, knowing a touchdown would send them through to the next round.
"We didn't want to leave anything in the bag. We had a chance to go down and score and win, and that's what we were trying to do," he said.
On the throw to Adam Thielen that set up the touchdown, Zimmer added: "It's a play we've used before – it's a great throw by Kirk and a great catch by Adam. You've got to take your shots when you're trying to win a playoff game."
Tight end Rudolph secured a four-yard pass from quarterback Kirk Cousins to cap a game-winning drive to open the additional period, silencing the majority of the crowd inside the Mercedes-Benz Superdome.
The victory means the Vikings – the sixth seeds in the NFC – will travel to San Francisco to face the 49ers next, aiming to cause a second shock result on their travels.
Once again in a postseason meeting between the teams, the Vikings found a way to get over the line despite squandering a 10-point lead in the fourth quarter.
Stefon Diggs scored on a 61-yard touchdown pass from Case Keenum to stun the Saints on the final play in January 2018, with the game later renamed the 'Minneapolis Miracle' due to the stunning finish.
This time Rudolph was the hero for Minnesota, though only after Cousins had set up the touchdown with a glorious 43-yard pass to Adam Thielen that moved the ball into the red zone.
Having also suffered a controversial exit at the hands of the Los Angeles Rams last year in the NFC Championship Game, New Orleans are left to come to terms with another heartbreaking play-off exit.
They had led 10-3 heading towards half-time, only to concede 10 points inside the final three minutes of the second quarter. Dan Bailey slotted over a field goal before the Vikings capitalised on a Drew Brees interception, Dalvin Cook going in from five yards out on the ground to make it 13-10.
The gap was extended further when the running back scored again in the third quarter, yet the Saints rallied. Brees connected with Taysom Hill for a 20-yard touchdown before Will Lutz levelled the score at 20-20, landing a 49-yard field goal with two seconds remaining in the fourth quarter. However, that would be the last time the home team had the ball.
After Thielen's catch, Cook failed twice to score on the ground from close range before Cousins went to the air again, picking out Rudolph in the back corner of the end zone.
Minnesota released veteran tight end Rudolph, who was selected in the 2011 NFL Draft, on Tuesday.
The 31-year-old was set for the second season of a four-year, $36million deal, but Rudolph is headed for free agency for the first time in his career as the Vikings save more than $5m in cap space.
"I got so lucky, because -- I didn't just get drafted by some team who 'had a need at tight end,' Rudolph wrote in a story published by The Players' Tribune.
"I didn't just get drafted as, like, the nameless, faceless '#1 tight end on the board.' I got drafted by a team that was all set in terms of need... but then drafted me anyway.
"I'll always remember that: how the Minnesota Vikings wanted me -- and wanted to bet on my potential."
In his time with the Vikings, Rudolph started 132 of 140 games, tallying 453 catches for 4,488 yards and 48 touchdowns – all those numbers in the top 10 in franchise history.
Rudolph's touchdowns are the most for a Vikings tight end in their history, while his catches and yards only trail Steve Jordan.
A Pro Bowler in 2012 and 2017, Rudolph had 28 receptions, 334 yards and a touchdown for the Vikings last season.
"Kyle has been a leader and mentor for us on and off the field from the first day I arrived in Minnesota," Vikings head coach Mike Zimmer said in a statement.
"He has been such an important part of this team and community throughout his career and it has been an honour to coach him the last seven seasons. He will be missed, and we wish him and his family nothing but the best."
