But Rubio did not handle the pressure well during the latest debate, said Weber, who has worked on campaigns for such Republicans as Mitt Romney and George W. Bush.
"I think the whole attitude of the Republicans toward the race changed a lot this weekend," he told CNBC's "Squawk Box."
Going into the weekend, most Republicans wanted to narrow the field, establish a clear challenger to Cruz and Trump, and wrap up the contest quickly, he said.
"People are hitting the brakes, saying let's not rush to judgment. Let's get this right. I think there's a greater emphasis on experience," he said.
Republicans now want to see how more experienced candidates fare in New Hampshire and South Carolina, Weber said. That means the Republican battle could last longer than thought last week, he added.
New Jersey Gov. Chris Christie launched a debate night attack challenging Rubio's record that news media focused on throughout the weekend. But the change in sentiment is most likely to benefit Ohio Gov. John Kasich and former Florida Gov. Bush, said Weber.
Rubio has the support of 17 percent of likely Republican voters in New Hampshire, putting him behind frontrunner Trump, who commands 30 percent of that group's support, according to a NBC News/Wall Street Journal poll conducted after the Iowa caucuses.
Cruz trails just behind Rubio with 15 percent, followed by Kasich with 10 percent and Bush with 9 percent.