Villa are one point behind Everton who drew 2-2 with Liverpool in a controversial Merseyside derby on Saturday which left Liverpool defender Virgil van Dijk with a serious knee injury that will require surgery.
While the Premier League goal rush slowed a little on Sunday, it did not in north London where Tottenham and West Ham battled out an extraordinary derby.
Son Heung-min put Tottenham ahead inside a minute and two goals from Harry Kane had Spurs three up after 16 minutes.
Bale, back at the club he left in 2013 to start a trophy-laden career at Real Madrid, came off the bench in the 72nd minute for his eagerly-awaited homecoming, but the script was not written for the Welsh winger.
Fabian Balbuena's 82nd minute header and Davinson Sanchez's own goal set up a nervy finale and then in the fourth minute of stoppage time Manuel Lanzini fired an unstoppable shot from outside the area to spark wild celebrations from the visitors.
Bale missed a great chance when the score was 3-2.
"That's three matches at home and two draws that could perfectly be two victories, and it's a big punishment because we look to the table and it's difficult to believe after scoring so many goals and playing so well that we only have eight points," Spurs manager Jose Mourinho said.
West Ham are the first team in Premier League history to avoid defeat in a game having trailed by at least three goals as late as the 81st minute.
"We are playing against a team everyone is talking about and after 10 minutes I was thinking 'is this going to be six or seven?' We changed and it made us better," West Ham manager David Moyes said.
Palace had hung on to the lead given them by Wilfried Zaha's 19th-minute penalty but three points would have been more than they deserved against their arch-rivals Brighton.
Substitute Alexis Mac Allister netted in the 90th minute to give Brighton a point although they had Lewis Dunk sent off in stoppage time for a poor tackle on Gary Cahill.
Sheffield United substitute Billy Sharp converted a late penalty to earn his side a point against Fulham who had taken the lead shortly before when Ademola Lookman marked his first start with an excellent solo effort.