Carlisle, the last team to win a league game at Elland Road, fell behind to a controversial goal by Jermaine Beckford on the half-hour mark.
Furious Carlisle manager Greg Abbott was sent to the stand by referee Russell Booth for a prolonged protest over a penalty award.
Booth ruled that Michael Doyle was just inside the box when fouled by central defender Richard Keogh as the midfielder met Luciano Becchio's knock-down.
Although goalkeeper Lenny Pidgeley dived to block Beckford's weak spot-kick, the striker followed up to stab home the rebound for his seventh goal of the season.
Abbott, a former Leeds academy coach, harangued the referee's assistant and the Carlisle boss was ordered to the stand two minutes after the goal.
Leeds winger Robert Snodgrass shot against a post moments before Abbott's dismissal, but four minutes after leaving the bench, Scottish striker Dobie climbed to head Kevan Hurst's corner past Casper Ankergren to deny the home side three more points.