Mr Gorsuch, who was sworn in by Justice Anthony Kennedy, becomes the 113th justice to serve on the high court.
"I am humbled by the trust placed in me today," he said after taking the oath.
"I will never forget that to whom much is given, much will be expected," the 49-year-old continued.
"And I promise you to do all my powers permit to be a faithful servant of the constitution and laws of this great nation."
Six key cases Gorsuch could rule on
Trinity Lutheran Church of Columbia v Comer - A Missouri church denied state funding for a playground in a case concerning separation of church and state.
Masterpiece Cakeshop v Colorado Civil Rights Commission - A Colorado baker who refused to make a wedding cake for a gay couple.
Peruta v San Diego - Does the second amendment grant California gun owners the right to carry a concealed weapon in public places?
North Carolina v North Carolina NAACP - A North Carolina voting overhaul that was said to target African Americans "with almost surgical precision".
Hernandez v Mesa - The case of a 15-year-old Mexican who was on the Mexican side of the border when he was shot dead in 2010 by a US border patrol agent.
President Trump's executive order banning travel from six Muslim-majority countries is also probably heading to the Supreme Court later this year.
Mr Gorsuch's appointment to the Supreme Court comes after a year-long political battle over filling the vacant seat.
Republican lawmakers refused to consider Barack Obama's nominee to replace Justice Antonin Scalia, a conservative darling who died in February last year.
It was the longest period a seat has remained unfilled on the Supreme Court since during the American Civil War in 1862.
On Monday morning, Chief Justice John Roberts administered Mr Gorsuch's first oath, which all federal employees take, at a private ceremony at the Supreme Court.
All eight justices, Mr Gorsuch's wife and two daughters and Maureen and Eugene Scalia, the widow and son of the justice Gorsuch is replacing, attended the ceremony.
Later at the White House, Justice Kennedy administered a second oath to Mr Gorsuch, who was his former law clerk.
At one point during Neil Gorsuch's Rose Garden swearing-in ceremony, Donald Trump thanked Senate Majority Leader Mitch McConnell "for all he did to make this achievement possible".
The thanks were certainly well earned. Mr McConnell quite possibly did more than any other US senator in history to ensure his party's nominee was confirmed to the Supreme Court.
He defied tradition in holding the late Justice Antonin Scalia's seat vacant for nearly a year so Democrat Barack Obama couldn't get his man on the court.
He then reversed more than half a century of precedent by abandoning the filibuster rule and allowing Mr Gorsuch a straight majority vote.
With Mr Gorsuch on the high court Mr Trump has fulfilled a campaign promise that was instrumental to his election last autumn.
The prospect of a liberal majority serving as the final arbiter of legal disputes was terrifying to many hard-core and evangelical conservatives who might have otherwise been reluctant to fall in line behind their party's unorthodox nominee.
Instead they voted for Mr Trump in droves - by a larger margin than they did Mitt Romney in 2012.
The deal is now done, and the rest of Mr Trump's presidency stretches before him. While the Gorsuch nomination unified Republicans of all stripes, the coming political battles will hardly be as clear-cut - and Mr McConnell may not be the hero for the president that he was today.
Mr Gorsuch, a former Denver appeals judge, was confirmed 54-45 on Friday after Senate Republicans took the historic step of changing the chamber's rules in order to ram through their pick.
On Thursday, Senate Majority Leader Mitch McConnell triggered a legislative manoeuvre known as the "nuclear option" when Republicans lacked the 60 votes required to end debate on Mr Gorsuch.
The result is a triumph for Mr Trump's young presidency. For many of those who voted for him, securing a conservative judge on America's highest court was a top priority.
