Sporting Kansas City's stalemate with leaders DC United on Saturday handed Jay Heaps's side an opportunity to leapfrog them in the standings if they could get past Columbus.
The hosts took a big step toward capitalising on that slip-up on 20 minutes when Lee Nguyen fired Revolution ahead.
It lasted less than five minutes, however, as Ethan Finlay struck shortly after to restore parity as both sides went in level at the break.
Both outfits cancelled each other out for much of the second half, but New England moved ahead again on 67 minutes through Jermaine Jones.
The prospect of protecting their lead improved dramatically when Columbus Crew were reduced to 10 men after Waylon Francis saw red for a second yellow card.
Indeed, the numerical advantage was enough to preserve the one-goal lead as Heaps's men held on to move four points behind United in the standings.