All-Australian forward Greiser, brought in during the off-season from St Kilda, kicked the decisive major with three minutes to play.
The Tigers then defied the hosts in a mad final minute with the ball deep in the Lions' forward 50.
Brisbane had an early three-goal buffer before the lead see-sawed in a gripping battle until Richmond prevailed 6.4 (40) to 5.4 (34).
The clash between two of last season's top-four teams was fierce if not fluent, the intensity slowly lifting and tackle pressure soaring in a tense final term.
With nine minutes left, Brisbane forward Dakota Davidson was bumped after taking a tight mark diving forward and received a 50-metre penalty.
That put her in the goal square and ensured the Lions earned a seven-point lead.
But Tigers skipper Katie Brennan won a marking contest right in front and again levelled the scores with four minutes to play before Greiser's big moment.
"How good's this girl? An absolute superstar and she's slotted seamlessly into our group," Brennan said of Greiser.
"What a win by our girls today."
Charlotte Mullins didn't have to wait long for her first AFLW goal, marking in front and kicking straight to open the Lions' account in the first quarter of her debut.
Brisbane had three goals before Brennan replied, the Tigers captain chipping from 50 metres and watching as the ball bounced truly and dribbled over the line.
Brennan then turned Brisbane defender Phoebe Monahan around with a clever kick, before Monique Conti breezed by and nailed her checkside shot at goal.
And when Emelia Yassir won a free kick and goaled the Tigers had levelled proceedings.
Courtney Jones then joined the party as the Tigers' fourth goal with 30 seconds until halftime pushing them ahead by six points to cap a dominant term.
Lion Sophie Conway had a shot at goal in the third term which missed to the right, but made up for it minutes later when using the breeze to snap a set shot from the boundary to tie the scores at the final break.
Conti starred with 26 disposals but had to be assisted off the ground late in the fourth quarter after early grabbing at her ankle.
"It was lost in the second quarter ... that was pretty disappointing; we looked like we had control," coach Craig Starcevich said.
"I don't think we handled it too well; there were some opportunities there to ... put the breaks on it."
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