Ian Henderson's first-half dismissal put the visitors on the back foot, and set up an open second period in which both sides had chances to snatch all three points.
It was Lillis, though, who was by far the more stretched of the goalkeepers and he kept the Shakers out to secure his side a valuable point which takes them to second in the League Two table.
Henderson was sent off for two bookable offences in the space of 10 minutes, firstly being cautioned for deliberate handball and then for pulling back Danny Mayor two minutes before the break.
After an opening 45 minutes during which there was only one shot on target, the second half was much more entertaining as Bury pushed forward and Dale set up to hit them on the counter-attack.
Shakers defender Pablo Mills scuffed a volley against the crossbar after referee Darren Drysdale missed Peter Vicenti's handball, while Michael Rose hit the woodwork at the other end with a bending free kick.
With one of the final kicks of the game, Jake Carroll lost his marker at a corner to head powerfully towards the roof of the net, only for Lillis to make a reflex stop and tip it over the crossbar.