Rangers 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.
Former Brentford manager Warburton 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 or 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."
Dumbarton boss Stevie Aitken was pleased with the effort put in by his part-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 a packed 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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