The 20-year-old Millwall frontman struck after 20 minutes and then again three minutes before the break in a remarkable first half at the Showgrounds in Sligo on Friday night.
Samir Carruthers had given Ireland a 17th-minute lead before O'Brien opened his account, and a third from Jack Grealish had the game all but won with just 25 minutes played.
But Gunnar Zachariasen pulled one back for the visitors within seconds, and the Faroes' hopes of mounting a fightback improved significantly when defender Ragnar Nattestad was fouled by Republic skipper Shane Duff inside the box and referee Alexandr Aliyev pointed to the spot.
Zachariasen duly obliged from 12 yards to make it 3-2 with eight minutes of the opening 45 still to play.
However, Noel King's men regained a measure of control within two minutes when Anthony Forde's first goal for the Under-21s gave them breathing space once again, and O'Brien's second sent them in at the break with their three-goal advantage re-established.
The second half proved to be something of an anti-climax by comparison as both sides had chances, but respective keepers Sean McDermott and Teitur Gestson escaped without further mishap.
Victory completed a double for the Irish over the Faroes and took their Group 6 points tally to seven from five games on a day when runaway leaders Germany needed a late equaliser in Montenegro to snatch a 1-1 draw as they surrendered their 100 per cent record.