The result of the final ballot from Tory MPs was announced by 1922 Committee Chairman Sir Graham Brady at 4pm today.
Mr Sunak and Ms Truss will now progress into the last stage, with Ms Mordaunt dropping out.
In the final round of voting by Tory MPs, Ms Mordaunt received 105 votes, Mr Sunak 137 and Ms Truss 113.
The new Tory leader will now be selected by the 200,000-strong party membership.
Liz Truss and Rishi Sunak
PA
The winner is expected to be confirmed on September 5.
Speaking after the results were announced, Ms Truss thanked supporters for entrusting her.
She tweeted: “Thank you for putting your trust in me.
“I’m ready to hit the ground from day one.”
While Mr Sunak’s campaign said he had secured a “clear mandate” from Tory MPs and will now “work night and day” to win the backing of the Conservative membership after he made it onto the final ballot for the leadership contest.
A statement read: “This is a really strong result with a clear mandate from MPs.
“He will now work night and day to get the mandate from the wider Conservative party family to beat Labour, protect the Union and seize the opportunities of Brexit.
“The choice for members is very simple: who is the best person to beat Labour at the next election? The evidence shows that’s Rishi.”
In a statement after her elimination from the contest, Ms Mordaunt congratulated Mr Sunak and Ms Truss.
Penny Mordaunt has dropped out of the running
Aaron Chown
“I pay tribute to anyone who puts themselves forward for such a demanding role,” she said.
“Politics isn’t easy. It can be a divisive and difficult place. We must all now work together to unify our party and focus on the job that needs to be done.
“I am a One Nation, proud Brexiteer. My campaign put forward a positive vision for the country I love so much, remembering who we are here to serve.
“Our mission is not only to deliver on what we promised but to win the fight against Labour at the next general election. I hope to play my part in both.”
Boris Johnson, after resigning earlier this month, appeared in his final PMQs session earlier today.
In it, he said his successor will "wipe the floor" with Labour leader Sir Keir Starmer, comparing them to “some household detergent”.
The Prime Minister also gave some “words of advice” to his successor during his final Commons appearance, which included: “Cut taxes and deregulate wherever you can and make this the greatest place to live and invest, which it is.
“I love the Treasury but remember that if we’d always listened to the Treasury we wouldn’t have built the M25 or the Channel Tunnel.”