The Jambos took the lead after a shaky start through Bruno Aguiar and despite being pegged back twice in a thrilling second period they won 3-2 courtesy of Roman Bednar’s late clincher.
Rix had envisaged a difficult afternoon with Tynecastle legend John Robertson taking charge of Livingston for the first time at home, but felt they just deserved the victory.
He said: “It was a tough game. They’re fighting for their lives and Robbo had them fired up. But I think overall we probably deserved the three points.”
Rix told his side how important victory was after Rangers, Hibernian and Kilmarnock all won the day before as they chase the Jambos for the second Champions League spot.
“After yesterday’s results we needed to win today,” he said.
“We made the players aware of that and they understood that. We had good occasions to put the game beyond Livingston and they’ve scored two goals out of nothing and we’ve still found enough to get a victory. I take my hat off to the lads for keeping going.”
Richard Brittain levelled with a free-kick from the flank which evaded everyone and - after Edgar Jankauskas had restored Hearts’ lead - Dave MacKay equalised with a 20-yard volley. Rix said: “The first one, you see that just about every week - where there’s a big inswinging ball and people run across it - it’s difficult to put the blame on anyone. The second one was smashed into the top corner.
“I just think overall we were probably the most composed side. It wasn’t a free-flowing performance but we dug in and matched Livingston and we have got to do that to grind out a result.”
But Robertson felt his side did not deserve to lose their ninth match in succession, a result which leaves them seven points adrift at the bottom.
: McKenzie, McNamee (Snodgrass 78), Strong, Dorado, Mackay, Brittain, Adams, Walker, Dair, Morrow, Dorrans (Hoolahan 70).
: Gordon, Neilson, Berra, Webster, Fyssas, Cesnauskis (Beslija 86), Petras, Aguiar, Skacel (Hackett 82), Jankauskas (Bednar 75), Elliot.
: J Underhill.