McCarthy (left) scored his second goal of the season
Hamilton picked up their first SPL point since September after recording a draw at home to Falkirk.
James McCarthy volleyed the home side into a deserved lead after 36 minutes with a volley from the edge of the box.
Substitute Michael Higdon equalised for the Bairns early in the second half when he headed past Tomas Cerny.
McCarthy came close again in the second half before Falkirk's Scott Arfield saw his last-minute shot blocked by goalkeeper Tomas Cerny.
The result means Hamilton remain bottom of the Scottish Premier League, two points behind Falkirk.
McCarthy had a chance to put Hamilton in front in the opening minutes when he met Lucas Akins' cross with a shot which went off target.
Goalkeeper Scott Flinders touched Akins' shot onto the post before the striker missed an excellent chance to give Accies the lead.
Left-back Brian Easton carried the ball from the half-way line into the penalty area before crossing towards Akins, who got the ball caught under his feet before turning his shot wide.
At the other end, Burton O'Brien's shot after the half-hour rebounded off the Hamilton defence as the visitors struggled to create openings.
Hamilton were in front nine minutes before half-time when Falkirk failed to clear Easton's throw-in and the ball fell to McCarthy, who lashed into the net past Flinders.
Patrick Cregg and Alex Neil played their part at New Douglas Park
Falkirk defender Darren Barr headed over before Hamilton captain Alex Neil was off-target with a free-kick effort in injury time.
Midfielder Patrick Cregg was replaced at half-time by Higdon and the striker made an instant impact four minutes into the second period.
O'Brien's cross was flicked on by Scott Arfield and Steve Lovell before Higdon got the vital contact that brought Falkirk level.
A 30-year effort from O'Brien flew over the bar before, at the other end, Flinders' attempted clearance cannoned back off Mark McLaughlin and drifted narrowly wide.
Flinders then tipped McCarthy's powerful drive over the bar.
Barr had a free header from Neil McCann's corner, but his diving effort was off target.
Hamilton's Derek Lyle crossed for fellow substitute Joel Thomas, who could only nudge the ball into the arms of Flinders.
Gerard Aafjes fired wide from McCann's cross and David Graham and Mark McLaughlin came close for the home side in the closing stages.
The final effort of the match saw Cerny block from Arfield as both sides were forced to settle for a point.
Hamilton Accies boss Billy Reid: "We're happy to stop the rot, so to speak, but I think we deserved to win the game over the 90 minutes.
"I think their goalkeeper made more saves than ours and we put numerous crosses into the box without getting a break.
"But I've got no complaints; the players worked really hard, you could see how much they wanted it and I feel for them a bit."
Falkirk manager John Hughes: "I thought Hamilton wanted it a bit more in the first half but we improved after the break and could have gone on to win it.
"My boys showed plenty of character to get back into it and I have to applaud that because Hamilton had some big lads in there.
"I've seen my players do better but it's a point gained for us and it keeps our unbeaten run going.
"I'm still looking for a 90-minute performance, it's been fits and starts for us all season."
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