The left-back's shot was blocked by Danny Simpson but Cabaye moved onto the loose ball to score his second goal in two games.
However, a narrow loss to Everton was followed by defeat at Atletico Madrid, whom they face in the Champions League quarter-final return leg on Tuesday.
After that comical error, Palace looked to pile the pressure on, with Wilfred Zaha and Andros Townsend looking particularly unsafe down the flanks. The Eagles are now seven points clear of the relegation zone with a tally of 35. On reflection, when you finish a game with a point having been 2-0 down, you have to be incredibly satisfied with that point however precious it might be moving forward.
The Eagles were up against it after goals in either half from Robert Huth and Jamie Vardy, but Yohan Cabaye quickly cut the deficit and Christian Benteke's 70th-minute header secured a share of the spoils.
Allardyce has attributed his side's recent run of results to much improved defensive displays but would not have been happy when Palace fell behind to a routine long-throw routine after just six minutes. I got the chance to work with people like Wayne Rooney who comes under a lot of stick but I saw at close quarters what a professional he is, and a leader he is and a great player he is. It was the Belgian who had leapt above the centre-back to meet the excellent Andros Townsend's delivery and power home Palace's equaliser, though City were outraged the goal was not disallowed with the forward "riding", according to Shakespeare, on the defender's back.
"It's a tremendous fightback from the players and that wouldn't have happened when I first arrived - it just shows the change of mood of the players", Allardyce was quoted as saying by the BBC.More news: Pence says US commitment to South Korea stronger than ever
- Leicester City have picked up more points under Craig Shakespeare in his three Premier League away games (4) than in the previous 13 managed by Claudio Ranieri (3). "Teams can go on a run, as we've proved, so we need to keep that momentum going".
"We have played against Madrid, then four days later Leicester, now we are going to play Osasuna, three days after that we are going to play in England".
Which Premier League club has had the most permanent managers since 2006/07?
Palace travel to Liverpool on Sunday, 23 April (16:30 BST), before another tough encounter, hosting Tottenham on Wednesday, 26 April (20:00 BST).
As if betting on the English Premier League was not hard enough, the final weeks of the season always throw a different set of imponderables into the mix, and one that could play a big part this weekend is the European factor in Crystal Palace against Leicester City and Manchester United against Chelsea.