Let me narrow down "engaged Buddhism" to "applying the Dharma to social and political injustices".
In the Kathavatthu Sutta, the Buddha did not prefer monks to engage in discussions concerning mundane things:
Then the Blessed One ... addressed the monks: "For what topic of
conversation are you gathered together here? ..."
"Just now, lord, after the meal, ... we ... engaged in ... conversation about kings, robbers, &
ministers of state; armies, alarms, & battles; food & drink; clothing,
furniture, garlands, & scents; relatives; vehicles; villages, towns,
cities, the countryside; women & heroes; the gossip of the street &
the well; tales of the dead; tales of diversity, the creation of the
world & of the sea; talk of whether things exist or not."
"It isn't right, monks, that sons of good families, on having gone
forth out of faith from home to the homeless life, should get engaged
in such topics of conversation ....
"There are these ten topics of [proper] conversation. Which ten? Talk
on modesty, on contentment, on seclusion, on non-entanglement, on
arousing persistence, on virtue, on concentration, on discernment, on
release, and on the knowledge & vision of release.
At the same time, the Buddha advised the lay followers in the Sigalovada Sutta on how good friends should be. From here, I extrapolate that lay Buddhists can become good friends of their fellow members of society, and exhort them to be likewise. This can be one way to Engaged Buddhism.
"Young man, be aware of these four good-hearted friends: the helper, the friend who endures in good times and bad, the mentor, and
the compassionate friend.
"The helper can be identified by four things: by protecting you when you are vulnerable, and likewise your wealth, being a refuge when
you are afraid, and in various tasks providing double what is
"The enduring friend can be identified by four things: by telling you secrets, guarding your own secrets closely, not abandoning you in
misfortune, and even dying for you.
"The mentor can be identified by four things: by restraining you from wrongdoing, guiding you towards good actions, telling you what
you ought to know, and showing you the path to heaven.
"The compassionate friend can be identified by four things: by not rejoicing in your misfortune, delighting in your good fortune,
preventing others from speaking ill of you, and encouraging others who
praise your good qualities."