The New York Yankees have the most games by a team when scoring 20+ runs, with 31 games.

Interpreted as:

team most regular-season games with 20+ runs, all-time

	TEAM	G	R	W%	W	L	AB	H	2B	3B	HR	RBI	BB	HBP	SO	SB	CS	PA	TB	XBH	SH	SF	IBB	GIDP	AVG	OBP	SLG	OPS
1	Yankees	31	651	1.000	31	0	1,419	653	124	33	107	619	246	22	146	30	0	1,711	1,164	264	15	9	6	5	.460	.543	.820	1.363
2	Reds	18	398	1.000	18	0	851	404	66	20	63		128	16	80	17	3	1,015	699	149	15	5	7	3	.475	.548	.821	1.369
3	Dodgers	18	381	1.000	18	0	796	378	58	17	52	354	124	13	66	20	1	954	626	127	16	5	1	10	.475	.549	.786	1.335
4	Red Sox	18	399	1.000	18	0	826	397	87	14	58	381	156	13	83	15	1	1,007	686	159	3	9	4	12	.481	.564	.831	1.394
5	Cubs	17	371	.941	16	1	764	362	67	12	56	348	133	10	69	15	1	923	621	135	11	5	4	13	.474	.554	.813	1.367
6	Giants	17	374	1.000	17	0	838	402	79	25	42	318	96	7	59	25	1	960	657	146	17	2	0	2	.480	.536	.784	1.320
7	Twins	16	330	1.000	16	0	714	338	59	14	36		122	9	48	26	1	870	533	109	15	10	4	6	.473	.549	.746	1.295
8	Athletics	15	320	1.000	15	0	694	346	58	16	47	306	96	9	66	20	0	813	577	121	10	4	1	5	.499	.562	.831	1.393
9	Indians	14	314	1.000	14	0	661	317	55	12	33	294	105	2	47	23	2	790	495	100	21	1	0	8	.480	.551	.749	1.300
10	Phillies	11	237	.909	10	1	498	236	64	10	35		85	7	63	16	0	611	425	109	14	7	5	4	.474	.549	.853	1.403
11	White Sox	11	233	1.000	11	0	503	238	52	13	26	215	64	10	50	6	2	596	394	91	12	7	6	2	.473	.534	.783	1.318
12	Braves	10	213	1.000	10	0	456	218	46	11	35		66	6	59	10	2	541	391	92	7	6	0	4	.478	.543	.857	1.401
13	Tigers	10	204	1.000	10	0	452	218	46	11	14	189	65	7	41	23	1	536	328	71	12	0	1	0	.482	.553	.726	1.279
14	Cardinals	9	195	1.000	9	0	430	192	38	10	17	172	64	5	31	11	0	510	301	65	11	0	3	0	.447	.523	.700	1.223
15	Angels	7	152	1.000	7	0	347	167	31	7	26	142	35	7	30	4	1	394	290	64	1	4	1	9	.481	.532	.836	1.368
16	Pirates	7	150	1.000	7	0	334	160	27	11	19		43	5	20	12	0	389	266	57	6	1	1	1	.479	.543	.796	1.339
17	Rangers	6	136	1.000	6	0	283	142	27	3	25	133	37	3	33	2	1	329	250	55	0	6	1	5	.502	.553	.883	1.437
18	Blue Jays	6	134	1.000	6	0	295	142	30	4	26	132	33	7	48	2	0	340	258	60	0	5	0	1	.481	.535	.875	1.410
19	Orioles	5	108	1.000	5	0	239	112	25	2	13	95	32	3	15	4	0	278	180	40	3	1	0	2	.469	.535	.753	1.288
20	Astros	4	86	1.000	4	0	194	91	27	1	16	79	21	1	29	2	0	219	168	44	1	2	1	4	.469	.518	.866	1.384
21	Rockies	4	80	1.000	4	0	177	80	17	1	16	79	17	5	28	0	0	205	147	34	2	4	1	1	.452	.502	.831	1.333
22	Padres	4	85	1.000	4	0	193	85	14	2	17	82	29	2	25	5	0	228	154	33	2	2	0	0	.440	.513	.798	1.311
23	Brewers	4	82	1.000	4	0	193	93	17	1	11	80	29	1	26	10	0	227	145	29	1	3	0	1	.482	.544	.751	1.296
24	Royals	4	90	1.000	4	0	193	93	21	2	8	86	26	1	22	3	0	223	142	31	0	3	0	0	.482	.538	.736	1.274
25	Mariners	3	64	1.000	3	0	136	61	16	1	13	62	25	2	22	4	1	167	118	30	1	3	0	0	.449	.530	.868	1.398