Murray was playing in his first Davis Cup match in two years and underlined his importance as Britain overcame Croatia 4-1 in their play-off encounter on the clay of the Stadion Stella Maris in Umag.
Despite nursing a back injury, the Scot clinched the all-important third point his team needed to ensure their berth in the elite World Group for the first time in five years when he eased to a 6-4, 6-2, 6-4 victory over world No 35 Dodig on yesterday.
Tougher tests await but the Wimbledon champion has plenty of confidence in the team, captained by Leon Smith, and wants to play his part, fitness permitting.
“I feel we’re ready to do well,” Murray said last night. “We have a top doubles team.
“Dan [Evans] and James [Ward] will continue to improve, get their rankings up a little bit between now and the end of the year, and we’ve got the makings of a very solid team.
“If I’m fit and healthy I’ll be there to play.”
Britain held the advantage going into yesterday’s reverse singles after Murray and Colin Fleming won in the doubles 24 hours earlier against Dodig and Mate Pavic.
There were slight worries over Murray’s back injury, which had left him a doubt for Saturday’s doubles match, but he dispelled any doubts with a clinical performance against Dodig.
With the world No 3 at the forefront of his team’s charge, few countries will relish the prospect of facing Britain.
Murray is unfazed at their next opponents, although he admitted he was keen on a home encounter.
“I don’t really mind [who we face next],” he added. “There’s certain teams you’d like to avoid but there’s teams we’d have a good chance of winning against.
“I just want a home tie. To get back in the World Group, get a big arena and play against one of the top teams at home would be good.”
Despite Olympic, US Open and Wimbledon glory in the last 13 months, Murray admitted he felt the pressure to perform well before facing Dodig.
He explained: “I was nervous. I wanted to try to finish the tie, it was a big match for us, we wanted to try to get back into the World Group, the guys have had some big wins this year and I didn’t want to let the team down.
“I’ve dealt with enough pressure around Wimbledon time and those sorts of matches to know how to get through it and how to conquer the nerves and fears that you have before you go out on the court.
“I’m glad I managed to finish it off because he fought extremely hard.”
Murray’s success in the first match of the reverse singles yesterday ensured the fifth tie, which was initially slated to be Evans against 16-year-old Borna Coric, was a dead rubber.
However, world No 149 Evans put the gloss on the scoreline with a professional 6-4, 7-6 (7/4) win over late replacement Pavic to finish a memorable week for Britain in style.
GB captain Smith was effusive in his praise for the entire team. He said: “It’s been a long journey a real team effort and couldn’t be done without all the back room staff.
“I’m delighted, really delighted. Team spirit has got us back and that makes the difference.”