The damage was all done in the first half of Wednesday's Premier League clash as the Hammers held off a spirited Seasiders team, who replied through Charlie Adam, in an entertaining contest at Bloomfield Road.
Neal Eardley hit the post and Marlon Harewood forced Robert Green to make a fine save as the hosts dominated after the break but the Hammers clung on.
Keane's flying start after joining the club on loan from Tottenham and the weekend and the form of Obinna, who scored a hat-trick on his previous outing, offered much hope for manager Avram Grant.
Grant made five changes to the side which beat Nottingham Forest in the FA Cup at the weekend and the result was a vibrant first-half display.
Blackpool were buoyed by the presence of their captain Adam after his widely-predicted deadline-day move to Liverpool failed to materialise.
And the Scotland midfielder immediately tried to dispel any suggestions that the club's refusal to sell him might affect his game with a powerful run in the opening minute.
Midfield partner David Vaughan also looked keen to influence proceedings as the Seasiders, with new signing Andy Reid on debut, played their usual attacking game.
But it was West Ham who had the better of the opportunities in a lively opening.
Richard Kingson was twice called into action early on, firstly to save a close-range shot from Obinna after Keane's scuffed effort diverted a Frederic Piquionne cross into his path.
The Ghana goalkeeper then plucked a long-range strike from Gary O'Neil, who was making his full Hammers debut, out of the air with little difficulty.
The visitors had another chance when Mark Noble's drive rebounded to Parker but the Hammers talisman fired well wide.
Kingson was nearly caught out when Craig Cathcart almost turned a Bridge shot into his own net but he made a quick reaction save to his right.
The Tangerines keeper was at fault, however, as West Ham claimed the lead after 24 minutes.
Kingson looked to have Obinna's ambitious shot from wide on the left covered but let it slip from his grasp and roll over the line.
It was a timely moment to score as it soon became apparent West Ham had injury problems.
During one stoppage in play James Tomkins, Wayne Bridge and Danny Gabbidon all went down in apparent pain.
Gabbidon was unable to continue and was replaced by Winston Reid but, despite obvious discomfort, Tomkins limped on while Bridge quickly recovered.
Reid went close to an equaliser after a poor clearance from Tomkins fell to him outside the area but Green brilliantly palmed over his long-range effort.
West Ham doubled their lead on 37 minutes as Keane made his mark.
Blackpool got themselves into a mess at the back as Noble lobbed a free-kick into the area after Parker was felled.
Kingson twice saved from Obinna in quick succession but got no help from his defenders and Keane's sliced effort found the net before it could be cleared.
Blackpool immediately responded as Adam found the back of the net direct from a corner, Frederic Piquionne completely missing his kick at the near post and unsighting Green.
Yet the Tangerines' joy was short-lived as Obinna restored a two-goal advantage on the stroke of half-time with a ferocious shot from 20 yards.
The game tightened up in the second half as West Ham looked to deny Blackpool any further way of getting back into the game.
The midfield battle intensified and after Noble and Adam collided, Parker was booked for a late challenge on Vaughan.
Blackpool manager Ian Holloway decided to go for all-out attack on the hour, sending on two more strikers in new signing James Beattie and former West Ham forward Harewood.
Beattie, who joined the club on loan from Rangers on Monday, had a chance moments after coming on but failed to make firm contact and Green easily gathered.
Neal Eardley went close to pulling one back but his 20-yard free-kick struck the post and West Ham scrambled clear.
Blackpool kept up the pressure with Adam flashing a shot narrowly wide and Reid making a vital clearance.
A good run by DJ Campbell caused further alarm but a fluffed backheel by Gary Taylor-Fletcher ended a promising attack.
Harewood forced Green into a fine save in injury time but the Hammers celebrated a crucial win.