Victory at Turf Moor put Jackett's men within seven points of the top-six as they hampered the hosts' own hopes of making up ground on Nottingham Forest.
A brace from Paul Robinson put the Lions on course before Andros Townsend netted with three minutes left to cap off a fine display.
"It keeps the play-offs open to us and that is an achievement after coming up from League One," said a buoyed Jackett. "We have nothing to lose - we have to keep believing - but a high number of points is required.
"If we attack it like we did in this game, and with the same appetite, then we have to keep believing we have a chance. From our point of view now we want to make sure everybody stays fit and this type of week can galvanise you.
"Our away record recently hasn't been as good as it should be - but we haven't been able to get over that line."
Source: PA
Source: PA