Rory Best was the other try scorer in slippery conditions in New Plymouth while Jonathan Sexton added a conversion and a penalty before being replaced by Ronan O'Gara early in the second half.
O'Gara sent over the extras to Bowe's second try before Paul Emerick claimed a consolation score for the States right at the death.
"We didn't play well," was O'Driscoll's assessment of the patchy performance ahead of next Saturday's huge clash with the Wallabies.
"We had to fight for our win. I suppose we got the win but we have a hell of a lot to improve on.
"We couldn't get any continuity and we were a little bit loose.
"It's good to get 80 minutes under your belt, though. Now we've got a big challenge at the weekend against Australia."
The victory was Ireland's first in five games following four straight defeats to Scotland, England and France (twice) in the recent World Cup warm ups.
But it was hardly surprising that the manner of the success didn't impress O'Driscoll as the Irish struggled for fluency against a nation they were expected to beat comfortably.
There were far too many handling errors even making allowances for the poor weather conditions and Sexton's kicking was another cause for concern as he slotted just two from six at the Taranaki Stadium.
Ireland's failure to pick up a bonus point leaves them second in the group, just behind Australia after their next opponents secured a maximum-point win over Italy earlier in the day.