Starting from the basics: Law 18.1 says that the batsmen have scored a run:
A run is scored [...] so often as the batsmen, at any time while the ball is in play, have crossed and made good their ground from end to end.
Law 18.9 states
When a batsman is dismissed [...] No other runs shall be credited to the batting side, except as follows. If a batsman is
(b) dismissed Run out, the batting side shall also score runs completed before the wicket was put down.
So in this circumstance, the batsmen completed one (and only one) run before the wicket was put down; therefore the batting team scores one run and the scores are equal. We now refer to Law 21.5:
The result of a match shall be a Tie when the scores are equal at the conclusion of play, but only if the side batting last has completed its innings.
Assuming that you are referring to a limited overs match, the side batting last has completed its innings, and the scores are equal. Therefore the match is a tie.
If this were the final ball of a timed match, then the match would be a draw as the side batting last had not completed its innings (except in the case that the run out was the last wicket; then it would again be a tie).