While their former club Birmingham City fly high in eighth place, their new purchase languishes just one point above the drop-zone after a dour game that saw the Hammers thanking goalkeeper Robert Green and the woodwork for saving them from defeat.
Sam Allardyce had made just one switch from the team that had climbed into 11th spot with victory over Wigan Athletic on Wednesday evening as Christopher Samba returned in place of substitute David Hoilett.
Following his 16th-placed side's back-to-back draws at Aston Villa and Portsmouth, Gianfranco Zola named an unchanged side from the side that had earned a point against Pompey in midweek.
With both teams lining up in a 4-5-1 formation, the midfield was far too congested for either side to make any meaningful impression during a mediocre opening that did little to warm the 33,093 frozen, frustrated fans, while both keepers were thankful for their gloves as they looked on as redundant spectators.
On the half-hour mark, though, Paul Robinson was finally called into action, when he was forced to fist out Alessandro Diamanti's right-wing free-kick from his near post and, at the other end, the back-pedalling Robert Green clawed Morten Gamst Pedersen's inswinging corner over his right-hand angle.
Then, three minutes before the break, Pedersen whipped a 20-yard free-kick onto the Hammers crossbar to leave West Ham walking off at the break fortunate to still to be level at the interval.
With the second period going much the same way as the first, both managers made a switch as the hour mark approached, as top-scorer Carlton Cole replaced Radoslav Kovac and Brett Emerton made way for Keith Andrews.
Midway through the half, with West Ham now moving to a 4-4-2 formation, James Tomkins saw his shot deflected over and, after Scott Parker replaced Valon Behrami, Rovers brought on fresh legs up front with Jason Roberts taking on the responsibility for the lone strike role from Nikola Kalinic.
Those switches certainly brought the game to life as Green brilliantly palmed Pedersen's rasping 20-yarder around his left-hand post and, from the subsequent corner, Cole cleared Gael Givet's shot off the line.
With 15 minutes remaining, Roberts brilliantly outwitted Tomkins but, with the whites of Green's eyes in his sights, the exposed Hammers keeper denied the Rovers substitute with a superb low save.
At the other end, Diamanti saw Robinson in equally defiant mood as he tipped over his long-range effort and the Italian found himself thwarted once more when his low 20-yarder flew back through the pack and into the ribs of the visitors well-placed keeper to leave both sides settling for a point.