The London club also rallied from a goal down when prevailing 2-1 at West Brom on Tuesday and moved up to third in the table as a result of Sunday's success at Upton Park.
West Ham fell behind to a Wilfried Bony strike after 19 minutes but two superb headers from Andy Carroll either side of the break put the hosts ahead.
Swansea goalkeeper Lukasz Fabianski was then sent off for impeding Diafra Sakho on the break and the Senegal striker capped off a fine win with a late third.
Allardyce felt his side were good value for the victory, which ensured they moved above Southampton - who play fifth-placed Manchester United on Monday.
"That shows the character of our team now, we didn't feel as though falling behind was going to knock us back and it was the same at West Brom," Allardyce said on Sky Sports.
"We started to get better and better again and once Andy scored his first goal we were on a roll again. With the amount of chances we created today, three was about right, I think."
Allardyce is refusing to get carried away over his team's lofty position but is excited about the challenges ahead.
"The big thing was we put distance between us and the teams that were below us today," he added. "It gives us an opportunity to build from here.
"I think we had a full squad, apart from Mark Noble, and that squad is staying together.
"We'll need it over the Christmas period, when we've got Chelsea and Arsenal in two days.
"So it's about what can we gain up to there and how can we do against those two - that will tell us a lot about where we are going to finish this year."
Allardyce was delighted to see Carroll putting his injury problems behind him to score his first goals since March.
"If he continues to gain match fitness, he'll be unstoppable," said the manager.
"We know how good he is at heading the ball and now we're working on his movement, which Teddy (attacking coach Sheringham) has been talking to him about."