Johnson won with double the amount of votes than Hunt, with 92,153 votes versus 46,656.
He said it was an “extraordinary honour and privilege” to be elected Tory leader and will officially become Prime Minister tomorrow.
Johnson praised Theresa May for her "extraordinary service to this party and this country", saying it was a privilege to serve in her Cabinet.
He added: "I know that there will be people around the place who will question the wisdom of your decision.
NEW PM: Boris Johnson has achieved his dream (Pic: GETTY)
"There may even be some people here who still wonder what they have done."
He concluded his victory speech by telling the assembled Tory members: "And I say to all the doubters, we are going to energise the country.
"We are going to get Brexit done on October 31.
"We are going to take advantage of all the opportunities that it will bring in a new spirit of can-do.
"And we are once again going to believe in ourselves and what we can achieve.
"And like some slumbering giant, we are going to rise and ping off the guy ropes of self-doubt and negativity, with better education, better infrastructure, more police, fantastic full-fibre broadband sprouting in every household – we are going to unite this amazing country and we are going to take it forward.
"I thank you all very much for the incredible honour that you have just done me.
"I will work flat out from now on with my team that I will build, I hope in the next few days, to repay your confidence, but in the meantime the campaign is over and the work begins."
Boris Johnson (Pic: SWNS)
BoJo got a ringing endorsement from Donald Trump, who tweeted: “Congratulations to Boris Johnson on becoming the new Prime Minister of the United Kingdom. He will be great!”
Mrs May was bought a £1,500 designer handbag as a leaving present after Tory cabinet members had a whip-around organised by Michael Gove, Times reporter Steven Swinford tweeted.
Leaving Number 10, Rory Stewart said of Mrs May: "She was very, very dignified."
Earlier, sources close to Johnson reportedly said he will want to immediately demote leadership rival Jeremy Hunt, the current foreign secretary, in a cabinet reshuffle.
It will not be smooth sailing for Johnson, with the Brexit situation described as a “sticky wicket” by a political scientist.
Politics professor John Curtice from Strathclyde University said no PM has inherited a party as low in the polls before.
"They are inheriting what at the moment at least is a very sticky, very volatile wicket,” he added.
Johnson is 5/1 odds with Ladbrokes to be the shortest-serving Prime Minister ever.
Education minister Anne Milton has already resigned over Brexit, saying that she has “grave concerns about leaving the EU without a deal”.
EU Commission Brexit negotiator Michel Barnier tweeted: "We look forward to working constructively w/ PM @BorisJohnson when he takes office, to facilitate the ratification of the Withdrawal Agreement and achieve an orderly Brexit.
DISTANT SECOND: Jeremy Hunt pictured arriving with his wife Lucia this morning (Pic: AFP)
“We are ready also to rework the agreed Declaration on a new partnership in line with EUCO guidelines."
Runner-up Jeremy Hunt said Boris Johnson would make a "great prime minister" and conceded he had had a "big mountain to climb" in the contest.
"I never felt it was too much of an uphill struggle but I always knew that it was going to be a big mountain to climb," the Foreign Secretary told the BBC.
"And we sensed quite a lot of movement coming in our direction during the course of the hustings – from young people, from people the further you got out of London and the south east, the warmer the response that we were getting but we never knew what the baseline was.
"I think Boris fought a very, very good campaign. It was a very clear campaign, his messages were absolutely clear and I think, as I say, he will now be a great prime minister.
"It's a challenging moment for our country – this is perhaps the most difficult legacy that a prime minister has inherited in peacetime. It's going to be very challenging.
"But he has got the confidence and the optimism and the energy to get us through this and that's why I think he will be a great prime minister."