Cole sprang the offside trap and was only denied by a good Hennessey block, and in the aftermath, Jody Craddock cleared a goal-bound Dyer effort off the line.
Herita Ilunga then had a header tipped over and Dyer went close with a back-heeled effort blocked by the keeper as Wolves narrowly avoided going into the break 2-0 down.
Zola encouraged by 'excellent' Hammers
In the opening seconds of the second half, it was Wolves's turn to lay siege to the opposing goal.
Milijas saw a fierce shot tipped over by Green, and from the resulting corner Craddock drew a fine one-handed block from the England goalkeeper, before the Serbian debutant went close with another decent effort from distance.
Wolves' Kevin Foley cleared a Jack Collison effort off the line as the game continued at a relentless pace, and when captain Upson's simple header made it 2-0 in the 69th minute, the game seemed out of Wolves' grasp.
In injury-time, Green underlined his international class with another fine save from substitute Richard Stearman.
Wolves manager Mick McCarthy:
"We began well in the second half but the England keeper-elect kept them in it for a while.
"Overall it was a good performance, but it matters not, we've been beaten. As many opportunities as we had in the game, we had to take one and unfortunately for us, they did take theirs.
"We're not a Premier League team, we're a group of players who won the Championship and earned the right to be here - not many of them have played in the Premier League before."
West Ham manager Gianfranco Zola:
"I think it was excellent, we won away from home and played a good game. There's a a few things we need to improve but for a first game, it was very encouraging.
"We knew the beginning was going to be tough, and also the first five minutes of the second half, but the team prepared well and faced danger.
"They came out with some good stuff, in the end it could have been more but they (Wolves) gave everything so they kept us close to them."
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