Frank Lampard's side suffered successive Premier League home defeats against Bournemouth and Southampton before beating Burnley 3-0 on Saturday.
Abraham notched his 13th league goal of the season to put Chelsea two up against the Clarets in a dominant display by the hosts.
It was a welcome result for Chelsea in front of their own fans after several disappointments in west London this term.
Lampard had questioned whether Chelsea had the "balls" to break down defensive opponents, but Abraham says the way they ripped apart Burnley proved they can be entertaining and successful.
"We do have to do that. The fans want to come and watch beautiful football and recently in our home games we haven't been giving them that," Abraham said when asked if Chelsea felt a responsibility to excite their fans.
"You could see that we played beautifully against Burnley though and the fans were behind us.
"I think we had been coming to Stamford Bridge recently and it was just a bit too slow. Against Burnley we showed that we started sharp and it continued throughout the game."
Arsenal, Manchester United and Tottenham are all up next for the Blues at the Bridge, leaving Abraham and company well aware they must not suffer a drop in standards.
"We've got to make the stadium rock. We've got tough games coming here but I am always confident, especially after the Burnley win, that we can go out and enjoy ourselves," Abraham said.
"For me I was always confident, it was just a matter of time. We knew we had to give a performance against Burnley though and we went out full of confidence.
"It was a bit disappointing after the win at Arsenal to draw at Brighton but we had to take the confidence from Arsenal into this game."