That method of rolling percentile dice actually pre-dates D&D.
Reading percentile dice in this manner pre-dates D&D, and has been used consistently in the rules throughout editions of the game.
Percentile dice date back to at least 1963, when they were used in wargames by the US Naval War College to simulate percentage chances using a 20-sided die marked 0-9 twice, and rolling that two times to generate two digits.
These dice were adopted by wargaming hobbyists around 1971, with an early advertisement describing them as able to "throw numbers 1-100".
Gygax used them in Original D&D (1974) as actual twenty-sided dice, read 1-20. It was common to color half the numbers with a crayon, presumably reading one 0 as 10 and the other as 20. They were also used in their original function as percentile dice, as in the Monsters & Treasure rulebook where magic item charts clearly place a result of 00 after 99 (e.g. Shield +3 on a roll of 98-00), i.e. 00 is used as if it represents 100 rather than 0.
The idea of reading a die as 1-10 wasn't even introduced until Greyhawk (1975). Spindle-shaped d10s were even newer—the AD&D 1e Dungeon Masters Guide (1979), p. 10, expects players to use the d20 as a d10, and describes a new non-platonic actual d10 marked 0-9. The modern d10 actually marked 1-10, and the d10 marked 10-90 for percentile use, are much newer inventions.
Consistent use of 00 = 100 throughout D&D
AD&D 1st edition Players Handbook (1978), p.9, under "Strength", explicitly states that a roll of 00 is intended to mean 100:
Furthermore, fighters with an 18 strength are entitled to roll percentile dice in order to generate a random number between 01 and 00 (100) to determine exceptional strength ...
The AD&D 2nd edition revised Player's Handbook (1995), released after Gygax left the company, actually calls a roll of 00 "100" in random treasure/monster charts to avoid confusing new players. In the chapter "The Real Basics", p. 11:
When the rules say to roll "percentile dice" or "d100", you need to generate a random number from 1 to 100. One way to do this is to roll two 10-sided dice of different colors. Before you roll, designate one die as the tens place and other as the ones place. Rolling them together enables you to generate a number from 1 to 100 (a result of "0" on both dice is read as "00" or "100").
The D&D 3.5 Player's Handbook, p. 5, gives a similar definition, this time accounting for the existence of newer ten-sided dice marked 00-90 for specific use as percentile dice in D&D and percentile-based RPGs:
Two 0s represents 100. Some percentile show the tens digit in tens (00, 10, 20, etc.) and the ones digit in ones (0, 1, 2, etc.). In this case, a roll of 70 and 1 is 71, and a 00 and 0 is 100.
The D&D 4e Player's Handbook, p.8, accounts for dice actually marked 1-10 (despite the artwork showing a die marked 0-9):
You can use d10s to roll percentages if you ever need to. Roll 1d10 for the "tens" and 1d10 for the "ones" to generate a number between 1 and 100. Two 10s is 100, but otherwise a 10 on the tens die counts as a 0–so a 10 on the tens die and a 7 on the ones die is a result of 7 (not 107!).
And the D&D 5e Player's Handbook, p.6:
... Two 0s represent 100. Some ten-sided dice are numbered in tens (00, 10, 20, and so on), making it easier to distinguish the tens digit from the ones digit. In this case, a roll of 70 and 1 is 71, and 00 and 0 is 100.