The visitors from the Ukraine grew in confidence but the dismissal of Joonas Tamm, who saw his second yellow card for repeated foul play at the hour mark, changed proceedings on the pitch.
Falcao, 34, has has been in impressive form for Galatasaray so far this season, netting three times in the opening two rounds of the Turkish Super Lig.
"This is a new level of escalation," team manager Rouven Schroeder commented after coach Achim Beierlorzer unsuccessfully asked the team to start training.
The Wolves took a hard-earned win and are now one victory away from booking their berth in the Europa League's group stage, as ten-man Desna were unable to cause surprise.
"We know that nobody can ignore the importance of Paolo, but we have to try replace him and the national team has responded," Gareca told reporters in a virtual press conference.
The teams had been due to meet at Estadio Azteca in Mexico City on September 30, but Costa Rican federation Fedefutbol said it had no choice but to withdraw from the match because of quarantine rules.
Matthew Wolff of the United States came second at even par and South Africa's Louis Oosthuizen was third on 282.
Potential successors for Wagner don't seem to be queuing up at the outfit's headquarters, but there may well be a new coach installed if the run of defeats continues.
Saturday ends with Southampton facing a tricky visit to play Burnley. Southampton have lost both league games so far and were dumped out of the Carabao Cup by Brentford, while Burnley lost their only league game away to Leicester City, before bouncing back to beat Milwall in the Carabao Cup. They have a good record against Southampton with four wins and three draws against them since October 2016.
In 1979, Bayern Munich players refused to accept Max Merkel as their new coach and threatened to go on strike.
Crystal Palace took the lead in the seventh minute through Andros Townsend and doubled their advantage through Zaha's retaken spot kick after David de Gea was judged to have moved too soon when denying Andrew Ayew after a handball.