The game turned at the start of the second half when Jackson scored the first eight points for New Mexico (8-10, 3-2 Mountain West) followed by a 3-pointer from Anthony Mathis for an 11-0 run that put the Lobos up 38-32. New Mexico was able to hold the lead the rest of the way, despite turning it over 20 times.
"He made some big shots early in the second half when we were still trying to find our rhythm offensively," New Mexico coach Paul Weir said of Jackson. "And he made some tough shots that I thought opened up the game a little bit and allowed our offense to flow from there."
Justin James scored a career-high 31 for the Cowboys (11-6, 2-2).
"James was obviously incredible (Wednesday)," Weir said. "That was as good an individual player in the conference that I've seen yet with his performance. He really picked up the slack. Going into the game we were really trying to limit (Hayden) Daulton. James hurt us tremendously."
New Mexico got scoring from just five players, but four of them finished in double digits with Furstinger scoring 17 with 10 boards and Makuach Maluach adding 17 points.
Wyoming has yet to win on the road in conference, dropping to 0-2. The Cowboys have lost five straight road games overall.
New Mexico's win keeps it perfect at home during conference play at 3-0 and moves it into fourth place in the Mountain West. It's the Lobos' first win over a team with a winning record since beating Evansville Nov. 29.
"It was our best win of the season. I really don't think there's much question about that," Weir said. "I think this team is finally buying into and seeing that you can win with emotion. You can win with passion. You can win with toughness. You can win with work ethic."
New Mexico entered the game with the conference's best turnover margin at plus-5. But while the Lobos surpassed their 12.6 average in the first half with 13 and finished the game with a minus-2 margin, they only had seven turnovers in the second half.
"We had just as many turnovers as deflections at halftime which is unfathomable to me," Weir said. "Thirteen deflections and 13 turnovers at halftime, which is completely unacceptable.
Wyoming, the worst offensive rebounding team in the conference, finished with just four off the offensive glass. And the Cowboys failed to make a 3-pointer in the second after going 4 of 8 before halftime.
James showed little wear from taking a knee to the groin area in Wyoming's previous game, a win over Boise State, scoring the Cowboys first seven points. Forward Jordan Naughton missed his third straight game with an injured back.
New Mexico leading scorer Sam Logwood missed his second game with an arm injury. Guard Troy Simons, who missed the last game with a conference-mandated suspension for twice being ejected for double technicals, also sat out Wednesday at Weir's discretion, sitting in the stands. There was no update to the status of either player moving forward, Weir said.
Wyoming is next at home Saturday against Colorado State.
New Mexico next travels to Fresno State on Saturday.
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Texas junior guard Andrew Jones is a limited participant in practice as he continues to regain strength following his diagnosis of leukemia, coach Shaka Smart said Monday.
Four-star point guard Jalen Gaffney made a verbal commitment to UConn on Monday, giving the Huskies their second ESPN 100 guard in the past two weeks.