Orica, which was the overwhelming stage favourite, clocked a time of 24 minutes, 42 seconds along the 21.7-kilometre course, which was lined by more than 140,000 fans.
The Australian team benefited from starting second and avoiding most of the rain that fell later.
"What an amazing day, I never in my life would have imagined something like this," said Tuft, a native of Langley, B.C. "I'm so proud of my team and I just can't thank them enough for that. They gave me the gift, it was really a birthday present. This team is really selfless that way. I feel really fortunate to be given that gift on my birthday."
Omega Pharma-QuickStep finished second, five seconds behind Orica, beating third-place BMC by just two seconds. Cadel Evans looked strong for BMC and is one of the favourites after last year's third-place finish.
With defending champion Vincenzo Nibali not competing, Tour de France runner-up Nairo Quintana is another of the race favourites and his Movistar team finished eighth, 55 seconds behind Orica.
One of his rivals, Joaquim Rodriguez, had a poor start as his Katusha team struggled throughout and only finished 19th. Rodriguez is 1 minute, 33 seconds behind Tuft and 38 seconds behind Quintana.
While it has become a tradition for the first Giro stage to be held outside Italy, this is the first time that the Grande Partenza — or the Big Start — took place outside mainland Europe.
Despite the wind and rain, it was a festive atmosphere in the Northern Irish city, with large crowds lining the route and cheering the riders on.
However, Irish rider Dan Martin crashed after just 15 minutes, leaving him with a suspected broken collarbone.
Martin caused the crash when he caught his wheel on a drain cover and went down strongly on his right arm, bringing down several of his Garmin-Sharp teammates.
The four cyclists who were able to continue had to wait for teammate Fabian Wegmann — who had been dropped early on — to catch up with them, as rules stipulate a team has to finish with at least five riders for its time to count. Garmin eventually finished last, 3:26 behind Orica. That means Ryder Hesjedal's chances of repeating his 2012 Giro success are all but over.
"Obviously there was a crash, and there was only four of us that stayed ahead," said Victoria native Hesjedal. "It was right before a corner and you hear something had happened, we were already through the corner, looked back and there was nobody there.
"Obviously it was pretty scary because you don't know what the hell is going on. I don't know what happened, it was basically a nightmare."
Belfast is also the start and finish point for Saturday's 218-kilometre leg, which takes in picturesque rock formation, the Giant's Causeway along its coastal route.
The Giro ends in Trieste on June 1.