West Ham were left counting the cost of a contentious decision by referee Uriah Rennie after drawing at Newcastle.
Cole (left) and Solano (right) both ended up on the scoresheet
Carlton Cole's volley and an easy turn and shot by Marlon Harewood gave West Ham a shock 2-0 lead before James Milner's disputed goal in injury time.
The referee let it stand despite Scott Parker being flagged off-side, before Nolberto Solano equalised from the penalty spot after the break.
Substitute keeper Steve Harper did well to deny Harewood a last-gasp winner.
There was no sign of the drama to come in a quiet opening period, before West Ham, who had not managed a goal away from home since August, scored twice in five minutes.
The opener came when debutant Calum Davenport nodded Yossi Benayoun's right-wing corner back to Cole to volley home from close range.
Within five minutes Harewood doubled the lead as he shrugged off the featherweight challenge of Peter Ramage to divert George McCartney's long low pass into the bottom corner.
Nigel Quashie and Nigel Reo-Coker were key to West Ham's forward momentum, but the early goals spurred Newcastle into action.
606 DEBATE: Give your thoughts on the match
First, Martins shot over then Alan O'Brien rattled the post with a low effort before Milner's controversial goal.
His low cross-cum-shot from the right wing went into the bottom corner after going through the legs of Parker, who was flagged off-side, but the referee overruled his assistant and let the goal stand.
Newcastle drew level after the break when Boa Morte conceded a penalty, converted by Solano, for handling a Milner free-kick.
Davenport had a header ruled out for climbing on Paul Huntington, before Newcastle goalkeeper Shay Given limped off with a groin injury, to be replaced by Steve Harper.
Martins came within inches of putting Newcastle ahead with a diving header from Solano's shot against the post as the home side gained the upper hand in an increasing frantic game.
Substitute Matty Etherington's indecision in the box cost West Ham the chance of a breakaway third goal as Newcastle's defence struggled to adapt to the new goalkeeper.
With Newcastle looking the more likely to score, home fans' cries for a second penalty were rejected when McCartney was adjudged to have controlled the ball with his chest.
Harewood had a chance to snatch a late winner for West Ham, but Harper did well to deny him a rebound from a close-range effort.
Newcastle manager Glenn Roeder
"When you consider we were two down and hadn't started well, to come back as we did was pleasing, especially after the FA Cup defeat on Wednesday.
"The second half was excellent and we probably let them off the hook.
"It's worth noting that When we finished, that was the 20th different back four we've had this year."
West Ham manager Alan Curbishley on Scott Parker being off-side for Newcastle's first goal:
"He let the shot go through his legs. If wasn't there Roy Carroll would have had a clear sight of it.
"If we'd come in at 2-0 it would have been a different second half.
"If we were midtable the off-side would have been given and the disallowed third goal would have stood, but when you're at the bottom, you don't."
Newcastle: Given (Harper 67), Solano, Ramage (Edgar 30), Huntington, Carr, Milner, Butt, Parker, O'Brien, Dyer, Martins. Subs Not Used: Luque, Pattison, Carroll.
Booked: Martins, Milner, Dyer.
Goals: Milner 45, Solano 53 pen.
West Ham: Carroll, Dailly, Ferdinand (Spector 86), Davenport, McCartney, Benayoun, Quashie, Reo-Coker, Boa Morte, Harewood, Cole (Etherington 73). Subs Not Used: Green, Sheringham, Mullins.
Booked: Carroll, Ferdinand, Dailly, Harewood.
Goals: Cole 18, Harewood 22.
Ref: U Rennie (S Yorkshire).