Martin hit a manhole on the Newtownards Road, just before Bridge End, and brought down three of his teammates on Friday evening.
He was rushed to a nearby hospital where doctors diagnosed a broken collar bone, which will put an end to his tour.
Meanwhile, Tuft praised the brotherhood within his Orica Greenedge team as the driving force behind his push to the Maglia Rosa.
In changeable conditions, Orica were the second team out and set a blistering pace as they pulled 22 seconds clear of the lead out team, Colombia, after just 7.9km.
They went on to clock a time of 24.42 for the 21.7km course before the Northern Irish rain caused chaos down the field, with 27-year-old Martin's Garmin-Sharp team colliding, and leaving a pre-race favourite's dream in tatters.
Martin came into the race in good form before his crash, while sprint specialist Marcel Kittel also hopped over a kerb in testing wet conditions.
As the last few teams headed out on the course, the roads improved and there were late attempts from Cadel Evans' BMC Racing team and Rigoberto Uran's Omega Pharma-Quickstep team to storm into the lead but ultimately, the dry conditions paid dividends for birthday boy Tuft.
The 37-year-old Canadian was in imperious form as his team 'gifted' him the Maglia Rosa.
He said: "We came here with great expectations. Our team is designed around the team time trial and we had high expectations.
"It was one of those things where the team manager said if we were on a good one Svein goes across the line first. So yeah, for me it was a very special day.
"It is really a dream come true for a guy like me. It is a once in a lifetime occurrence and I have to thank my team for the gift."
The crowds were truly impressive. I never expected to see that kind of a crowd.
Tuft paid homage to the thousands of fans who lined the course in Belfast, from the start line right through to the finish.
He added: "The entire course was lined four or five deep with people wearing pink and screaming. I never thought it would be like that."
Having spent two years at Garmin with Martin, Tuft also wished his rival well following his crash.
The Canadian said: "I just saw some footage and it looked pretty rough. Dan is a great kid and I rode with him on Garmin for two years.
"I feel bad for him that it happened in his homeland and in a race he could really do something at. I hope he is OK."
Tomorrow sees the riders take on a 219km stage which again begins at Titanic Quarter before heading along the coast, passing the Giants Causeway and Rathlin Island before returning to Belfast via Cushendall and Carrickfergus.
Martin's compatriots Nicolas Roche and Philip Deignan finished in fourth and fifth respectively as the much anticipated Big Start finally arrived in Northern Ireland.
Nicolas Roche said his ride was tougher than he expected.
He added: "I was expecting it to be a tough day and it happened to be an extremely tough one.
"We were looking at the forecast on the team bus and we were hoping the rain was going to hold off but at least now we can say it rained for everyone.
"At 20 seconds behind Orica it is a good time. Orica are probably the strongest team today and it was going to be hard to beat them.
"This morning I was saying that if we could be within 30 seconds of the winning team then it would be a good day."
He added: "We have no specialists in the team apart from Mick [Rogers] and I think it is one of my strong points, so it is a great result.
"There are two or three riders in the team that took on their first team time trial and they gave everything. They were brilliant today."
Team Sky's Philip Deignan added: "We were pretty unlucky with the wet roads and we couldn't take too many risks with the course so we lost time there but the team rode well.
"I won't forget the reception we got yesterday for a while, it was amazing.
"It was like a wall of noise but all I could make out was a lot of 'Phils' and 'Deigns' shouts. We lost a lot of time on the corners but the fact we are near the top shows what we could have done in dry conditions."
Giro d'Italia stage 1 results and team general classification:
1 Orica Greenedge 0:24:42
2 Omega Pharma-QuickStep 0:00:05
3 BMC 0:00:07
4 Tinkoff-Saxo 0:00:23
5 Team Sky 0:00:35
6 Astana 0:00:38
7 Cannondale 0:00:53
8 Movistar Team 0:00:55
9 Giant-Shimano 0:00:56
10 Ag2r-La Mondiale 0:00:58
11 Trek Factory Racing 0:01:00
12 Belkin 0:01:01
13 Neri Sottoli 0:01:01
14 Bardiani-CSF 0:01:07
15 Androni Giocattoli 0:01:14
16 FDJ.fr 0:01:18
17 Lampre-Merida 0:01:20
18 Colombia 0:01:22
19 Katusha 0:01:33
20 Lotto Belisol 0:01:34
21 Europcar 0:01:48
22 Garmin-Sharp 0:03:26