It took a 78th-minute try by Conrad Smith, conjured in the wake of a brave decision by Aaron Cruden, to separate the rivals in a fiercely contested opener at Eden Park.
England finished with heroes all over the pitch, among them the heavily maligned Freddie Burns and Kyle Eastmond, but Robshaw is unable to look beyond the result.
"We're extremely proud of all the effort and that we were able to come here an do that, but we can't be happy," the captain said.
"Because of the standards we set for each other now and the standards we play at, we're here to win a Test series and it's simple as that.
"We're 1-0 down but there are two Tests left and we see the bigger picture.
"We genuinely thought we could win here. There's no point stepping out on to the pitch if you don't believe that.
"That's the position we have got to with English rugby - we believe that.
"As a squad, we're in a good place, but it's one thing being in a good place and it's another thing winning. We will get better.
"There's one thing we have to do differently next week and that's win."
England were missing as many as 10 first-choice starters either through injury or the fixture congestion that robbed them of their Aviva Premiership finalists, but still the All Blacks were forced to dig deep to amass a 15th successive victory.
The Northampton and Saracens contingents watched from the stands as New Zealand's two-decade triumphant run at Eden Park was threatened in a performance that bodes well for the second and third Tests.
Head coach Stuart Lancaster is now faced with a selection conundrum with the superb Rob Webber, Geoff Parling, James Haskell, Ben Morgan, Burns and Eastmond worthy of retention when the series resumes in Dunedin, yet they might all be dropped.
England have won just two Tests in 13 matches on Kiwi soil, but there is a real sense they can add to that record.
"New Zealand will improve, but so will we. This was our first hit-out for a while as well. Everyone is going to get better," Robshaw said.
"We know we're going to get stronger and stronger every week. We're going to be hard on ourselves this week.
"We know certain things didn't go our way, but we were pretty evenly-matched out there."
Burns and Cruden traded penalties but it was the All Blacks fly-half who took the bold decision to take a quick tap penalty instead of going for goal when the score was tied 15-15.
His instincts proved correct as soon after Smith darted over in the right corner with England stretched by Marland Yarde's loss to the sin-bin.
"We got ourselves in some good positions at times, but we didn't quite finish them off," Robshaw said.
"They probably got on our line twice and took a chance and that was the difference.
"Going into that last 10 minutes we had enough to win but credit to them, that's what they're good at.
"They got a chance and they took it, but we always knew that about them. We experienced it against them at Twickenham last year.
"We were in positions to win, but it's only the result that matters and that went their way."
- Sports & Recreation
- Freddie Burns
- Aaron Cruden
- Kyle Eastmond
- Conrad Smith