There have only been 14 1-wicket victories, and none between 1923 and 1951 and then between 1951 and 1980.
Here is a list of all 1-wicket and 2-wicket victories as on 25 Aug 2019:
1) The first instance was in the Eng v Aus Test starting at the Oval on 11/08/1902, when England won.
The last wicket partnership was of 15 runs from 248/9 to 263/9 by GH Hirst (58) and W Rhodes (6). The story goes that they planned to get the runs in singles, but they did not have to do this.
2) SA v Eng at Johannesburg starting on 02/01/1906, won by SA. Partnership of 48 from 239 to 287 by AW Nourse (93) and captain/keeper PW Sherwell (22).
3) Aus v Eng at Melbourne starting on 01/01/1908, won by Eng. Partnership of 39 from 243 to 282 by SF Barnes (38) and A Fielder (18).
4) SA v Eng at Cape Town starting on 01/01/1923, won by Eng. Partnership of 5 from 168 to 173 by AS Kennedy (11) and GG Macaulay (1).
5) Aus v WI at Melbourne starting on 31/12/1951, won by Aus. Partnership of 38 from 222 to 260 by DT Ring (32) and WA Johnston (7).
6) NZ v WI at Dunedin starting on 08/02/1980, won by NZ. Partnership of 4 from 100 to 104 by GB Troup (7) and SL Boock (2).
7) Pak v Aus at Karachi starting on 28/09/1994, won by Pak. Partnership of 57 from 258 to 315 by Inzamam-ul-Haq (58) and Mushtaq Ahmed (20).
8) WI v Aus at Bridgetown starting on 26/03/1999, won by WI. Partnership of 9 from 302 to 311 by BC Lara (153) and CA Walsh (0).
9) WI v Pak at St John’s starting on 25/05/2000, won by WI. Partnership of 19 from 197 to 216 by JC Adams (48) and CA Walsh (4).
10) Pak v BD at Multan starting on 03/09/2003, won by Pak. Partnership of 5 from 257 to 262 by Inzamam-ul-Haq (138) and Yasir Ali (1).
11) SL v SA at Colombo (PSS) starting on 04/08/2006, won by SL. Partnership of 2 from 350 to 352 by MF Mahroof (29) and SL Malinga (1).
12) Ind v Aus at Mohali starting on 01/10/2010, won by Ind. Partnership of 11 from 205 to 216 by VVS Laxman (73) and PP Ojha (5).
13) SA v SL at Durban starting on 13/02/2019, won by SL. Partnership of 78 from 226 to 304 by MDKJ Perera (153) and MVT Fernando (6).
14) Eng v Aus at Leeds starting on 22/08/2019, won by Eng. Partnership of 76 from 286 to 362 by BA Stokes (135) and MJ Leach (1).
Scorecards of the first match in 1902 and the last match in 2019 are here and here.
We now look at the 17 Tests which ended in a two-wicket victory.
One of them was actually an one-wicket victory as one batsman had retired hurt and probably would not have batted. This was at Dunedin where NZ had pulled off an improbable win in 1980 (see No 6 above).
The 1985 match was NZ v Pak at Dunedin starting on 09/02/1985, won by NZ. The 9th wicket partnership (effectively the last wicket) put on 50 from 228/8 to 278/8 by JV Coney (111) and EJ Chatfield (21). Earlier BL Cairns (0) had retired hurt at 217/7 and the 8th wicket fell at 228.
Coming back to the list of 14 1-wicket victories:
The Feb 2019 match saw the highest winning 10th wicket partnership of 78 in Tests as well as in all first-class cricket. The previous Test record was 57 by Pakistan in 1994.
Later in Aug 2019 we had the second highest winning 10th wicket partnership of 76.
The highest scores in these partnership were 153 by Lara as well as Perera, followed by Inzaman’s 138 and Stokes’s 135.
The lowest scores by the no 11 batsman include 0 by Walsh in 1999, 1 by Leach in 2019 and several other single-figure scores. Walsh did slightly better in 2000 scoring 4.
Leach made his single run only towards the end of the partnership, which ensured the match would at least be tied when England’s score reached 358.
The highest score by a No 11 in these matches was 22 by Sherwell in 1906.
Yasir Ali was playing in his only Test.
Inzamam and Walsh are the only ones to appear twice in these partnerships.
The highest total here is 362 by England in 2019, and the lowest 104 by NZ in 1980.
That was an eventful match which saw DL Haynes scoring 55 and 105, being the only opener in all Tests to be last out in both innings. It ultimately led to New Zealand winning a series against West Indies. This was the first time they had won any Test series at home. They had earlier won only against Pakistan in Pakistan in 1969-70.
The August 2019 win by England also involved one of the lowest first-innings totals by a winning side (and the lowest such total since 1887):
However, this also meant that England had made its lowest score at Leeds: