After a poor first half, Albion took the lead when Jay Rodriguez bundled home a tap-in from two metres for his sixth league goal of the season in the 54th minute.
Swansea could ill afford to lose against such limited opposition and, after Andre Ayew missed an easy first-half chance by shooting wide, they grabbed a 75th-minute equaliser when Abraham rose to meet a cross from Sam Clucas.
The point leaves the Welsh side 15th, just above the relegation zone.
Abraham, who had not scored in the league since October, said Swansea were satisfied with the point against a team benefiting from the new manager "bounce".
"Every team that replaces their manager has a point to prove and that's what they did today," he said. "We got a point, and we'll keep playing for more, that's what matters."
Albion manager Moore faces the difficult job of trying to make up a 10-point deficit on 17th-placed Crystal Palace with five games remaining but he said he was pleased with the response he had seen after just three days with the players.
"I'm always one for if you don't win the game, you don't lose the game but I'm really pleased they showed character to stay in it, dig in and work well with each other," he said.
"I thought the players were first class."
(Reporting by Neil Robinson; Editing by Christian Radnedge and Clare Fallon)
