But Southampton were reduced to 10 men when Maya Yoshida was shown a straight red card for taking down Niasse in the 85th minute, and Everton challenged first in the 94th minute, when Leighton Baines' free kick was punched over the crossbar by McCarthy, before Davies' goal.
Though Everton have been safe for some time their manager's position is looking increasingly precarious.
"We'll be working again on that this week and going into the game next weekend with the same mentality we've had all season - to win".
Have any of our players broken into a sweat in that first half?
Today's visitors Southampton could be seen as a timely reminder of what might have been for Everton, with Saints heading into the final two games of the season in the bottom three.
Everton are eighth in the table having been 13th when Allardyce took over.
Substitute Nathan Redmond's first goal in nearly a year appeared to have put the fight very much back in Saints' hands as it would have left them needing a win at fellow strugglers Swansea on Tuesday to guarantee safety.
Jordan Pickford is one of the first names on the team sheet for Allardyce this season and the English worldwide will keep his place under the bar this weekend. Southampton have managed just one win in their last 15 trips away in the Premier League, losing eight times in the process.
To say the game was low on incident and excitement would be a considerable understatement. The school of science, it is being said by some unhappy supporters, is presently in special measures.
Both managers turned to their benches at the break as winger Redmond replaced the limping Mario Lemina, while Allardyce surprisingly called on Ramiro Funes Mori in place of Yannick Bolasie.
A half-time substitute, brought on at the same time Everton looked to be shutting up shop, Redmond added another dimension to Southampton's game to get at the opposition defence. Cue joy in the small corner of the Bullens Road Stand that was red and white and more boos from the blue majority.