This game took nearly 70 minutes to catch fire but when it finally did, it was Rovers who had the best chances to win it.
However, with Benni McCarthy and El Hadji Diouf both missing through injury, the home side lacked someone with the killer touch to make their dominance count.
Allardyce gave a debut to Pascal Chimbonda, who signed from Tottenham for an undisclosed fee earlier this week.
West Ham, meanwhile, stuck with the same line-up that started against Tottenham last weekend.
With both managers opting to start with a lone striker and a five-man midfield, it was no surprise that chances were few and far between in a cagey first half.
The tone for a disappointing opening 45 minutes was set when Jason Roberts could only find the side-netting when stretching to connect with a Chris Samba knockdown at the far post.
After that, it was the Hammers who posed the greater threat. Carlton Cole somehow wriggled past three defenders inside the area, but curled a shot beyond the far post.
Then the England international turned provider minutes later, laying the ball off intelligently for Luis Jimenez to pull the trigger, but Chimbonda averted the danger with a brilliant saving tackle.
The closest Rovers came to breaking the deadlock before the interval was a pot-shot from Samba, which was deflected over the crossbar.
The second half threatened to be just as uneventful as the first until the introduction of David Hoilett suddenly stirred Rovers into life.
The Canadian striker should perhaps have hit the target when he was played in by David Dunn but instead fired wide of the near post.
Then, ten minutes later, the speedy forward had an even better opportunity following more good work by Dunn, but his attempted chip drifted agonisingly over the crossbar.
Sensing West Ham were there for the taking, Allardyce threw on new £6million frontman Nikola Kalinic in the closing stages.
However, for all Rovers' late pressure, they could not muster a winner.