Within five minutes Chris Brunt had the game's first attempt on goal for the Baggies with a low shot from the edge of the box after being supplied by Matt Phillips, but it posed no problem for Joe Hart.
Gareth Barry headed just high and wide from a Phillips cross following a corner shortly afterwards, although as the half progressed it was then the turn of the visitors to apply some pressure down the other end.
Antonio began a move in his own half, running down the left and firing off a shot that Ben Foster was easily equal to.
Shortly afterwards Pedro Obiang tried his luck from 40 yards out, placing his audacious effort over Foster but into the crossbar. The Irons' offensive spell continued on the half-hour mark with a high half-volley by Carroll after an Antonio cross from the left, but the scoreline remained goalless.
Down the other end, Brunt and James Morrison worked together to orchestrate a promising chance for Phillips, but he struck just wide in the last meaningful chance of a first period with 10 attempts on goal but little real threat.
The visitors began the second half the brighter, as two minutes in Jose Fonte tried to head home from Aaron Cresswell's corner, but the attempt fell just wide.
Marko Arnautovic, returning from suspension, was brought on just before the hour mark as James Collins - who had needed treatment at the tail end of the first half - went off injured, prompting a reshuffle for the Hammers. The newcomer was in the wars himself minutes later with an aerial coming-together with Craig Dawson.
Minutes later there was another clash, resulting in a yellow card for Baggies keeper Foster which could easily have been a red. The Englishman rushed out of his box to intercept Javier Hernandez, taking him down with a high tackle but the referee opted to let him remain on the pitch.
The visitors provided the greater, though still mild, threat for the rest of the half, before Hal Robson-Kanu had a header on target easily saved by Hart a couple minutes before the end of normal time.
The result keeps both teams in the same positions they inhabited at the beginning of play, with the Baggies in ninth and West Ham down in 18th.