The home side will be particularly annoyed with themselves for the manner in which they conceded the first at the end of a half they dominated and were leading 13-6.
With just seconds left before the hooter was due to sound, Otago conceded a penalty inside their 22. Charlie Hodgson hoofed a well angled kick across field for Greenwood to collect and score - after two Otago defenders had allowed the ball to bounce.
The second score came when Jones took a lovely line to move onto a Shane Williams pass but once again the Otago defence was caught badly napping.
And defence was perhaps the biggest difference between the sides. Except for the lapse that allowed Danny Lee in for his 33rd minute try, the Lions guarded their defences much more zealously than Otago.
The home side were still well in it when Nick Evans landed his fifth successful kick, a 60th minute penalty, to leave his side trailing by just a single point (20-19). However, Hodgson cancelled that out four minutes later before Jones crashed over beside the posts for a try that really broke the hearts of this Otago side.
The Lions pack, much more assertive in the second half, were strenghtened by the arrival for the final quarter, of fresh legs in the considerable shape of Andrew Sheridan, Steve Thompson and Danny Grewcock (replaced Simon Shaw), and against a tiring Otago 8 they were able close out a game whose outcome was put beyond all doubt when the lively Shane Williams stepped inside the feeble tackle of Glen Horton to score a try that Hodgson converted.
Otago led 6-0 after Evans landed two penalties but Hodgson had the Lions level with two of his own before Otago scoed the game's opening try.
It came as a result of missed tackles by Denis Hickie and Hodgson on Nick Brew who then gave the scoring pass to Danny Lee.
At this stage Otago held the whip hand but they then suffered the crushing blow of conceding that try and it was the Lions who carried the spring in their step as they headed off for the break.
Evans pushed Otago in front with an early second half penalty but after Jones scored his try, the Lions took control up front and with the home pack falling away it was a question of them just trying to contain the Lions and limit the damage.
Ronan O'Gara came on for the final ten minutes and almost engineered two further tries with deft kicks that Simon Easterby and Geordan Murphy just failed to gather.
All in all a performance, after a shaky start, that will have pleased the Lions management and leave them in good heart for the Test next Saturday.