Cornyn, who headed bipartisan negotiations with Connecticut Senator Chris Murphy (D), said the legislation was a "positive and an affirming way" to help curb future mass shootings.
"I don't believe in doing nothing in the face of what we saw in Uvalde and we've seen in far too many communities," Cornyn said on the Senate Floor. "Doing nothing is an abdication of our responsibility as representatives of the American people here in the United States Senate."
The gun control bill passed by the Senate calls for strengthened background checks for prospective gun buyers under 21, criminal penalties for straw purchases and gun trafficking, clarification on the definition of Federally Licensed Firearms Dealers, and the end of the "boyfriend loophole."
It also provides $750 million in grants as incentivization for states to start crisis intervention programs and provides billions in federal funding to improve mental health services for children and families.
The legislation will now head to the Democrat-controlled House, where it's expected to pass despite Republican pushback.
"First thing tomorrow morning, the Rules Committee will meet to advance this life-saving legislation to the floor," Speaker of the House Nancy Pelosi said in a statement on Thursday.
Despite the bill not including all of the original measures that Democrats and the White House called for, President Joe Biden is expected to sign the bill, if it passes through the House.
Biden urged the House to "promptly vote on this bipartisan bill and send it to my desk" in a statement after Thursday's vote.
"Tonight, after 28 years of inaction, bipartisan members of Congress came together to heed the call of families across the country and passed legislation to address the scourge of gun violence in our communities," Biden said. "Families in Uvalde and Buffalo — and too many tragic shootings before — have demanded action. And tonight, we acted."
However, the bill is still opposed by House GOP leaders, who are pushing other House Republicans to vote against the gun control legislation because they believe it violates Americans' Second Amendment rights.
But Mitch McConnell, one of the 15 Republican Senators who voted for the bill on Thursday, said the legislation advances "commonsense solutions without rolling back rights for law-abiding citizens."