However just two minutes after the break Michael McIndoe's cross found Best who headed home to draw the Sky Blues level.
Clinton Morrison and Jose Fonte both had chances to win it for either side but the spoils were shared.
Crystal Palace went into the clash without a win in their last four visits to the Ricoh Arena but they started well with Ambrose firing over from 25 yards.
Minutes later Ambrose tried his luck with a free-kick from distance but Keiren Westwood made a comfortable save.
Ambrose was at the heart of Palace's attacks and with 20 minutes gone his cross found Johannes Ertl but Westwood made a superb fingertip stop to keep out the Austrian's header.
Palace continued to push forward and it was inevitable that it was the excellent Ambrose who put the Eagles in front eight minutes later.
The ball fell to on-loan West Ham striker Sears on the right and his drove in a low cross which was swept home from close range by Ambrose.
The home side gradually began to improve after going a goal down and McIndoe's cross resulted in a goalmouth scramble with Ertl eventually hacking the ball away.
With half-time approaching David Bell's shot was deflected out and from the resulting corner Aron Gunnarsson headed wide from close range when he perhaps should have done better.
The Sky Blues came flying out the traps after the break and got back on level terms just two minutes after the break when McIndoe did well down the left and whipped in cross for Best to head in.
Coventry immediately went in search of the lead and Bell unleashed a volley from the edge of the box which went wide.
However Palace eventually found their rhythm once again and Ambrose unleashed an effort which was well saved by the impressive Westwood.
The game opened up with both sides pushing forward looking for the crucial second goal and Bell's drilled shot was blocked by Shaun Derry.
Neil Warnock's side should have gone in front with 16 minutes remaining when Ambrose's corner found Fonte but he headed over the bar.
At the other end Morrison could have delivered a killer blow against his former side but Danny Butterfield got his body in front of the ball to make a crucial block.