A second-half goal from right-back James Tavernier took the Light Blues on to an uncatchable 79 points with four games remaining.
The Light Blues will now prepare for life in the Premiership although first they have next week’s Petrofac Cup final against Peterhead at Hampden followed by a William Hill Scottish Cup semi-final against old rivals Celtic at the national stadium, before they finish their league season.
The former Brentford manager said: “I am delighted for the fans for what they have been through, it’s great for the players and the club to get back in the top tier.
“It was a clear directive for the players this season but they have dealt with the weight of expectation and delivered. All credit to them.
“It is the first step and the worst thing we can do is stop moving forward.
“Enjoy tonight, enjoy the good times, we have a big game on Sunday, a packed Hampden with 95 per cent Rangers fans, hopefully we will perform well and win the trophy and play well the following week.
“We want to hit 91 points in the league.
“We want to win it in style, win it convincingly and get momentum for next season.
“We didn’t want to limp over the line, rely on someone else losing a game, we wanted to win it in style.
“We are clear now, we have four more games to play and we want to win them all.
“The pressure is always there at Rangers and if we drop our standards we will get hurt.”
Andy Halliday threw down the gauntlet to his title-winning team-mates and urged them to challenge Celtic next season.
In the post-match celebrations Halliday took a megaphone off one of the fans in the ultras-style section of the ground and started a song and afterwards he targeted a challenge to their traditional rivals.
The former Bradford City player said: “We’ve got to be challenging for the title; it’s as simple as that.
“And that’s not me being arrogant or disrespectful to Celtic at all, they are four in a row and going for another title, but you can’t be at a club like this and be happy with second.”
Asked what the fans had said when he went in among them, Halliday, who joined the Govan club last summer, replied: “Give us a song. Obviously I had to be careful which one I picked but I managed to grab the microphone and chanted away. I wasn’t in for too long because my throat’s not been great the last few days.
“It is an incredible achievement. Obviously it was our main goal since the start of the season.
“We feel we have been top of the season since day one of pre-season and have only been going one way and that is forward.
“Now it’s full steam ahead to the end of the season because we still have a lot of big games and then in to the Premiership, back where we belong.
“I think we are getting better and better.”
Tavernier told the club’s official website that it was “an unbelievable feeling” to score the winner and how he enjoyed the celebrations.
He said: “We pressed a lot and tried to create chances, but they defended really well and the keeper pulled off some great saves, so I don’t think the scoreline tells the story of the game.
“The fans have been great all season, both at home and away. It was a great feeling to walk around the pitch and be able to celebrate with them.
“There were probably more too who couldn’t come to the game as it was sold out, and it was just a fantastic moment with all the boys to walk round the pitch.”
Dumbarton boss Stevie Aitken was pleased with the effort put in by hispart-time players.
He said: “First of all we have to give credit to Mark Warburton and Davie Weir (assistant) and everyone at Rangers for winning the league.
“The pressure was on them this year after missing out last year.
“It was always going to be difficult for us coming to Ibrox.
“We’re a part-time side and some of my players finished 10-hour shifts at their work before the game.
“I listened to some of the radio phone-ins on Monday night and they predicted we’d lose by six or seven goals.
“For me that’s disrespectful to my players who, to a man, were outstanding here. They worked extremely hard.”
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