Carlisle led at half-time through Courtney Meppen-Walter's header and it needed a brilliant save from Vale goalkeeper Chris Neal to deny Liam Noble early in the second half.
But Ben Williamson and Tom Pope, with his 14th goal of the season, struck either side of the hour mark to turn the tables and end Vale's run of three straight defeats.
Adams said: "Chris' save was terrific and I was just about to make a triple substitution when Ben scored. I was going to make the changes to try to inject some life into us, so the goal came at a good time.
"I'm delighted both of our centre-forwards scored because they are judged on their return of goals. Tom's strike rate is good, but Ben's is not so good so that should give him a boost.
"Ben ran the channels really well in the second-half and he's got to understand the qualities he's got. Sometimes he doesn't quite do that.
"No-one has been more disappointed than me with how things had gone in recent games and the lads had got a bit down on themselves. Players can lose confidence and belief when a run of results go against them, so this should give them a lift.
"Results have gone for us (on Saturday) too and we've now got a nice buffer of 11 points above the relegation places. We should now start to look up the table again instead of down."