No question about it, looking into another human being's eyes creates a powerful connection. In fact, in the Western world avoiding eye contact is considered a sign of discomfort, even lack of veracity. (Be aware that in some cultures, direct eye contact is seen as overly aggressive, so you'll need to adjust your behavior accordingly.)
In a on-on-one conversation, practice maintaining eye contact so that your listener knows she has your full attention. In a large gathering, be careful not to employ the "group graze," making only brief eye contact with your listener while scanning the crowd, which implies that you're looking for people more important that she.
When talking in a group, slowly sweep from one side of the group to the other, making eye contact with each person in turn (do a mental count to five, if that helps). You'll find you can establish a meaningful connection with everyone present.