The teams were level at 2-2 at half-time, but the tourists were hit by two tries in six minutes to go 10 points behind and posed their only real threats in a frantic last 15 minutes.
Replacement hooker Daryl Clark went over for a try to give them hope of a first win in New Zealand since 1992 and winger Jermaine McGillvary touched down at the corner four minutes from the end only to have the score disallowed for a knock-on.
Despite the thrilling comeback, it was another disappointing performance from Wayne Bennett’s men, who were lacklustre in an opening 14-6 defeat by a Tongan Invitational XIII in Hamilton a week earlier.
The Lions will have a chance to square the series with the Kiwis in Christchurch next Saturday, but they will be haunted by the feeling that their hosts were there for the taking at Eden Park and that this was a glorious opportunity missed.
New Zealand coach Michael Maguire had responded to his team’s 26-4 defeat to Australia by dropping playmaker Shaun Johnson in favour of the equally experienced Kieran Foran, but his comeback lasted only five minutes before he went off with a dislocated shoulder.
And the Kiwis played the game without a specialist hooker after Maguire sprung a last-minute surprise by axing Brandon Smith for a breach of team protocol and bringing in half-back Kodi Nikorima, who was not even in the 24-man squad.
The Lions clearly heeded the message from Bennett as they kept their errors to a minimum - it was 34 minutes before they came up with a first handling error - and they had most of the possession in the first half.
The introduction of full-back Jonny Lomax added more thrust to the attack, while his St Helens team-mate Alex Walmsley made a big impact from the bench.
But there were too many tame endings to their sets and the only scores before half-time came from penalties by Jamayne Isaako (New Zealand) and Gareth Widdop (Great Britain).
The Kiwis produced an error-strewn first half but still managed to create the only tryscoring chances, with centre Joseph Manu failing to finish off a break by Benji Marshall and former Leeds winger Ryan Hall coming to the rescue by intercepting Jerome Hughes’ pass near his own line.
The breakthrough came eight minutes into the second half, when full-back Roger Tuivasa-Sheck wriggled free of Walmsley’s tackle to get Isaako racing away for the opening try.
Three minutes later, second rower Briton Nikora touched down a Marshall kick only to have the try disallowed for a knock-on in the build-up, but the reprieve was short-lived as substitute Corey Harawira-Naera used brute force to get the ball over the line despite the attention of four defenders.
Isaako kicked his second goal to open up a 10-point lead and Britain started to come up with the errors as they began to chase the game.
Clark brought them back into the contest on 68 minutes when he finished off a drive by Walmsley to score their only try and Widdop added the conversion before McGillvary’s close shave, when he lost his grip on the ball in Kenny Bromwich’s last-ditch tackle.