The Rams recorded a second successive win over the Christmas period at Pride Park on Saturday.
Ball, 19, struck his second senior goal from the edge of the penalty area in the second minute and midfielder Green, who returned from a long-term knee injury in November, headed his first goal in over a year eight minutes later.
Hammers striker Frank Nouble pulled one back for the visitors with a rasping drive just before the break, but Nigel Clough's side created other chances and defended valiantly in the second half to claim another prized scalp by following up their Boxing Day win over Leeds.
Derby have won three of their last four league games to climb within two points of the play-offs.
West Ham, depleted by injuries and suspensions, have won only one of their last five league games and could slip out of the top two if Middlesbrough beat Cardiff later this afternoon.
Derby were unchanged from the side that beat Leeds 1-0 on Boxing Day, while Hammers manager Sam Allardyce was without eight first-team players.
Skipper Kevin Nolan, winger Julien Faubert, defender George McCartney and midfielder Jack Collison all missed out through suspension, while Abdoulaye Faye (thigh), Guy Demel (thigh), Matt Taylor (calf) and David Bentley (knee) are currently injured.
Midfielder Gary O'Neil and striker Nouble both made their first league appearances of the season for the Hammers but Derby caught the visitors cold by racing into an early lead.
Striker Nathan Tyson slipped the ball inside from the right side of the penalty area and Ball rammed an unstoppable shot into the top corner from 18 yards.
The visitors had barely had time to catch breath when the Rams doubled their advantage.
Left-back Gareth Roberts stormed forward and supplied a fine cross for former Doncaster team-mate Green to head firmly downwards beyond West Ham keeper Robert Green into the bottom corner in the 10th minute.
West Ham midfielder Noble, skipper in Nolan's absence, lifted his side with a 25-yard shot that flew just wide and Papa Bouba Diop headed over the crossbar.
Hammers striker Frederic Piquionne also went close with a header and at the other end Derby striker Jamie Ward flashed two shots just off target.
Derby goalkeeper Frank Fielding saved brilliantly twice to deny Nouble and O'Neil in quick succession in the 39th minute, but could do nothing to prevent Nouble reducing the deficit three minutes later.
Piquionne played the ball inside from the left and Nouble arrowed a shot into the bottom corner from 20 yards.
Rams midfielder Craig Bryson fired wide early in the second half and Ball dragged his effort off target from similar distance inside the area.
Allardyce sent on strikers Sam Baldock and Carlton Cole with an hour gone, but it was Derby who threatened to add to their lead.
Green's goalbound header from John Brayford's cross was blocked by West Ham defender James Tomkins and Ward lashed a shot from the edge of the area wide.
The action continued to flow from end to end, with Noble and Cole this time pulling shots wide for the visitors.
Noble's late corner prompted Fielding into another sprawling save and Cole headed wide in the 90th minute, but Derby held on.