After a stodgy first-half showing, Brazil, who drew their opening match with Switzerland, wasted numerous chances before Coutinho (90+1) netted his second goal of the tournament to rescue his side.
Neymar (90+6) added gloss with a further goal in added time as Douglas Costa unselfishly squared the ball past goalkeeper Keylor Navas so Neymar could finish into an open goal.
The PSG star was at the heart of the game throughout, including winning a penalty which was sensationally overturned by VAR.
At the full-time whistle Neymar was in tears, presumably with relief that Brazil had avoided a damaging result in Group E.
Both these two nations play their final group game on Wednesday, with Brazil taking on Serbia and Costa Rica playing Switzerland.
Odds correct as of 1510 BST 22/06/18