First-half strikes from Lyle Taylor and Arron Taylor-Sinclair at Tynecastle moved the Jags 20 points above their opponents.
Partick finished with 10 men following the late dismissal of Stuart Bannigan, but they held on to record only their second victory in 17 matches which lifts them one point above Ross County into 10th place in the Scottish Premiership table.
"I would be very disappointed if we give that 20 points up to be honest," said Archibald.
"Hopefully today's victory will give us confidence and we can start looking at the teams above us, try and drag them into it.
"There are two or three about us and hopefully we can maybe make it interesting.
"I think it was vital we got the win. We had not won in such a long time, I think it was 10 games ago.
"With Ross County winning on Saturday as well, they've put a wee run together so it was vital we got a victory against Hearts."
While Partick are looking up the table, the future of Hearts looks bleak with Gary Locke's side, who were handed a 15-point penalty during the summer, now 19 points adrift of Ross County with 17 games remaining.
Locke said: "We have got to keep going, the mountain has just got bigger.
"We have got to keep working hard, that was the disappointing thing for me today, we can't just turn up for 45 minutes.
"It's extremely difficult. That was a big game. But we have got to keep going, there is nothing else we could do."