The Wimbledon champion, who beat 16-year-old Borna Coric on Friday and then teamed up with Colin Fleming to win Saturday's doubles, was far too strong for a weary Dodig who hung on grimly at the end, saving two match points before succumbing.
The Scot’s contribution was vital as Britain secured victory over a Croatia team weakened considerably by the absence of Marin Cilic and Ivo Karlovic in Umag.
Dan Evans then beat Mate Pavic 6-4 7-6(4) in the final rubber as Britain secured a 4-1 triumph.
"I thought it was a pretty good match from my side, I gave him very few opportunities," Murray told Eurosport. "I'm very happy with the way I played today.
"Davis Cup is different. The atmosphere was pretty hostile at the end there. But that's what's special about it. It's character building, you’re playing for your team and your country.
"Now we return [to the World Group], I'm looking forward to it. You get to play against some of the biggest countries, top teams, top players. It's exciting, I'm pleased for all the guys. I'll try and celebrate tonight with all the team."
Murray held his nerve despite playing in front of a partisan crowd who grew ever more vocal as the match wore on.
Team captain Leon Smith said: "It's amazing, it was an incredible performance. My voice has gone from all the shouting. The crowd was absolutely brilliant and from start to finish Andy was exceptional.
"It's been a long journey and a real team effort from everyone, all the back room staff. We're delighted, really delighted. Team spirit has got us back. It makes the difference, everyone gets behind each other after years of hard work."
Spain, inspired by the return of Rafael Nadal, needed only two days to crush Ukraine 3-0 to retain their place in the World Group, having been required to face a play-off after losing in the first round to Canada.
Switzerland and the Netherlands beat Ecuador and Austria respectively in their play-offs.
Later on Sunday in the semi-finals, 2010 champions Serbia must try to overturn a 2-1 deficit at home to Canada.
The winners will face the Czech Republic who cruised past Argentina in Prague with a day to spare.