The opening minutes saw little in the way of goalmouth action at either end of the pitch, with Spurs content to sit back and play on the break.
However, the visitors hit the front with 14 minutes on the clock as David Bentley released Gareth Bale down the left.
His cross was turned against the woodwork by Fraizer Campbell, but O'Hara was on hand to head the loose ball into the back of the net from the six-yard line.
Midway through the first half Spurs had under-fire goalkeeper Heurelho Gomes to thank as he made a smart block from Moestafa El Kabir's shot, before Chris Gunter crucially cleared the loose ball.
Shortly before the interval the visitors threatened to double their lead as O'Hara forced a smart, low save from home keeper Gabor Babos.
Soon after the restart Darren Bent went close to getting on the scoresheet, as his header was smartly turned over the crossbar by the alert Babos.
Bent and strike partner Campbell continued to combine well, making light of the absence of the cup-tied Roman Pavlyuchenko.
Twenty minutes from time Bent was replaced by in-form winger Aaron Lennon, leaving Campbell to plough a lone furrow up front.
Soon after El Kabir looked poised to grab an equaliser, but after finding space on the right side of the penalty area, he was denied by a superb last-ditch challenge from Spurs captain Jonathan Woodgate.
Late on the home side tried to up the pace, but Spurs held out with some ease to all but seal a place in the knockout stages of the competition.