Keenum reciprocated during his introductory press conference by saying the Broncos were his first choice.
"We got our guy; the guy that was our target," Elway said. "He was the perfect fit for us so we're extremely excited to welcome Case to Denver and really have high expectations of what we can do and what he's going to add to us, especially on the offensive side."
When Cousins was taken off the market by the Minnesota Vikings, the Broncos quickly reached agreement with free-agent Keenum on a two-year, $36 million deal.
After Elway spread praise on Keenum at the outset of Friday's press conference, he turned to Keenum and said he would like to introduce "the new starting quarterback of the Denver Broncos, Case Keesum, Keenum, I'm sorry."
Despite the general manager's slip, Keenum talked about his respect for Elway's playing career, saying he wore No. 7 (Elway's number) while with other teams because of Elway.
"Denver was No. 1 on my list, too," Keenum said.
He was not concerned about his current teammates lobbying for Cousins before the Broncos signed Keenum.
"Von [Miller, who publicly said the Broncos should sign Cousins] was the first person that reached out," Keenum said. "And I applaud those guys for wanting to make their team better.
"It's not awkward for me at all. Just the reception we've had, not just from the city, the fans, the coaches, but the players -- it's been great and I'm ready to step in that locker room and earn the right to be a leader."
"First of all, we want to be here long-term," Keenum said. "I want to continue to earn everything from this point out, so it made the most sense."
Keenum had his best season in 2017 after replacing injured Sam Bradford as the Vikings' quarterback in the second game of the season.
He completed 67.6 percent of his passes with 22 touchdowns and seven interceptions while going 11-3 in his 14 regular-season starts.
The Vikings won their first playoff game against the New Orleans Saints, but were beaten 38-7 by the Philadelphia Eagles in the NFC championship game. Keesum was 28 of 48 for 271 yards, one touchdown and two interceptions against the Eagles.
"I plan on being better than I was last year," Keenum said.