Behrens headed Union into the lead after just 52 seconds, the team's earliest goal in Bundesliga history, and added another just minutes later past a shellshocked Mainz defence.
Union goalkeeper Frederik Ronnow saved two Ajorque spot-kick penalties in the second half, snuffing out any chance of a Mainz fightback despite a goal from Anthony Caci.
Behrens headed in again for the home side midway through the second half, with Milos Pantovic adding a fourth late in injury time to seal the victory.
Behrens toasted his first top-flight hat-trick, telling DAZN "Until now, I've only scored one in a game".
"Simply outstanding, an incredible feeling - and we can only build on that."
"Craziness," said Ronnow, "to save two penalties (in a game), I've not done that before in the Bundesliga - I think not even one."
Mainz coach Bo Svensson said his side "rolled out the red carpet for Kevin Behrens" as Ajorque had a day to forget.
"To miss two penalties in a game, it's perplexing," Svensson added.
The glamour arrivals of Germany internationals Kevin Volland and Robin Gosens, the latter shattering the club's transfer record, dominated the lead-up but it was a typical Union signing in Behrens who struck early.
Plucked from Sandhausen in 2021, Germany's smallest professional football club, Behrens played his first top-division football at Union at the age of 30.
Behrens headed in a Jerome Roussillon cross from the left and then did the same from Aissa Laidouni on the right to double up after just nine minutes.
Union continued to look likely to score and it was not until the second half when Mainz found their fluidity, winning a penalty 61 minutes in.
Ajorque stepped up but Ronnow dove left, saving the effort.
The visitors got on the scoresheet just moments after thanks to a sweet volley from Caci but Union hit back almost immediately after, again through Behrens and again from a header.
Mainz had a chance of an unlikely comeback when VAR intervened to award a penalty for a handball with one minute remaining, but Ronnow again guessed correctly to save Ajorque's attempt.
Behrens was subbed off for Volland in the dying stages and the former Monaco man set up the final goal, curling in a cross for Pantovic to seal the victory for the home side.
Kolo Muani on the money
In Sunday's late game, France striker Randal Kolo Muani's first-half tap-in took Eintracht Frankfurt past derby rivals Darmstadt with a 1-0 win.
Kolo Muani, who has been heavily linked with a move to Paris Saint-Germain after a breakout debut season in Frankfurt, opened the scoring just before half-time.
With 15 goals and 11 league assists last term - the combined most of any player - Kolo Muani sidefooted in a Philipp Max pass to give his side the lead.
The visitors almost equalised early in the second half, but Frankfurt keeper Kevin Trapp got a toe to the ball.
Frankfurt held on as Darmstadt pushed for a late goal, giving new coach Dino Toppmoeller victory on his league debut.
Darmstadt returned to the top division for the first time since 2016/17 and just the fifth season overall.
