The Cobblers were good value for their narrow win heading into Sunday's second leg at Whaddon Road, but the Robins will feel more than capable of turning the tie around and returning to Wembley 12 months after their final defeat to Crewe.
O'Donovan's well-taken finish, his sixth goal of the season, came midway through the first half from Chris Hackett's right-wing cross.
Burly striker Adebayo Akinfenwa was denied a precious second goal for Northampton when he saw his close-range header kept out by a tremendous one-handed save by Scott Brown.
Paul Benson spurned two gilt-edged chances for Cheltenham while Jermaine McGlashan could feel hard done by not to have been awarded a second-half penalty when he was brought down, but the visitors failed to really trouble Lee Nicholls at Sixfields.
It was Northampton's first win in seven attempts over the Robins, who completed a league double this season, but the semi-final is far from over as both teams chase a return to England's third tier for the first time since 2009.
Cobblers boss Aidy Boothroyd was keen to stress pre-match that form counts for little heading into the play-offs, with his charges having ended a five-match winless streak on the final day of the regular season.
But even he could not deny that both sides have enjoyed starkly contrasting home and away form this term.
The Cobblers had won 11 of their previous 12 home games going into this one while the Robins are unbeaten in 20 at Whaddon Road, yet they had only managed 10 wins on the road between them.
With that in mind the onus was very much on Northampton to make the running tonight and it was they who settled first, with recalled pair Akinfenwa, twice, and Hackett off target.
Akinfenwa headed Ben Harding's 16th-minute corner goalwards but appeared to see his effort deflect off the arm of a Cheltenham shirt from close range.
And it was the hosts who took the lead in the 26th minute courtesy of a fine finish from O'Donovan, connecting with Hackett's cross from 12 yards.
Benson dragged wide a great chance to draw Cheltenham level immediately, and it was his header which forced Cobblers goalkeeper Nicholls into his first save in the 41st minute.
McGlashan made Nicholls tip over three minutes into the second half before Ishmel Demontagnac tested Brown at the other end while under pressure from Sido Jombati.
Benson then wasted another clear chance for Cheltenham just before the hour mark when he headed wide Marlon Pack's cross at the far post.
Cheltenham goalkeeper Brown produced a superb save to keep out Akinfenwa's 68th-minute header.
McGlashan could consider himself unlucky not to earn a penalty in the 70th minute having been clattered by Luke Guttridge.
It was Northampton who made all the pressing in the closing stages as they went in search of a crucial second but it failed to arrive and, despite a celebratory pitch invasion from the home support, the tie remains wide open ahead of the second leg.