Conference South outfit Thurrock joined the ranks of the FA Trophy's third round giant-killers by beating Aldershot Town 1-0 on Sunday.
Kris Lee got the only goal to knock out the Conference side from Hampshire.
Saturday's ties saw Forest Green, Dagenham & Redbridge and Scarborough all bow out to lower-ranked opposition.
And FA Cup heroes Exeter scored twice in the last six minutes to complete a recovery from 2-0 down to draw 2-2 at Ryman Premier side Billericay.
City, who play Manchester United in their FA Cup replay next week, scored through Jake Edwards and Sean Devine.
Nicky Bailey hit the goal as Conference leaders Barnet beat Farnborough 1-0.
Sunday, 16 January:
Carlisle United (2) 3 Preece 12, Livesey 43, Hawley 65
Redditch United (1) 1 Hollis 10
Att: 912 (at Workington FC)
Team Bath (1) 1 Jaukovic 8
Histon (0) 2 Cambridge 73, Nightingale 88
Att: 131
Thurrock (0) 1 Lee 57
Aldershot Town (0) 0
Att: 613
Saturday, 15 January:
Accrington Stanley (0) 0
Hereford United (0) 0
Att: 945
Barnet (0) 1 Bailey 71
Farnborough Town (0) 0
Att: 1,334
Barrow (1) 2 Tarrant 2, Knight 47
Scarborough (0) 1 Senior 50
Att: 1,076
Bath City (0) 0
Canvey Island (1) 3 Berquez 39, 56, Boylan 50
Att: 693
Billericay Town (1) 2 Trott 38, Forbes 46
Exeter City (0) 2 Edwards 85, Devine 86
Att: 1,380
Burton Albion (1) 3 Miller 39, Shaw 53, K Taylor 55
York City (0) 0
Att: 1,286
Crawley (0) 3 MacDonald 73, 79, Davidson 85
Worthing (2) 2 Holmes pen 29, Ekoku 45
Att: 1,343
Dagenham & Redbridge (0) 1 Uddin 90
Bishop's Stortford (2) 2 Morison 8, 21
Att: 931
Eastwood Town (1) 3 Knox 16, Sucharewycz 60, Mushambi 75
Spennymoor United (0) 2 T Raw 62, Oliver 63
Att: 263
Forest Green Rovers (0) 1 Powles og 64
Aylesbury United (1) 2 Cracknell 28, Gutzmore 63
Att: 554
Gravesend & Northfleet (0) 0
Eastbourne Borough (0) 0
Att: 862
Halifax Town (0) 0
Northwich Victoria (0) 1 Allan 75
Att: 1,345
Hednesford Town (1) 1 Barrow 39
Worcester City (0) 0
Att: 748
Hyde United (1) 3 Johnson 32, McNeil 49, Milligan pen 61
Whitby Town (2) 3 Williams 1, Gill 19, Nicholson 72
Att: 402
Kettering Town (0) 0
Alfreton Town (0) 0
Att: 1,113
Leigh RMI (0) 1 Reed 89
Altrincham (0) 2 Maddox 75, Little 86
Att: 401
Leyton (0) 0
Cheshunt (0) 0
Att: 84
Morecambe (0) 2 McFlynn 67, Curtis 88
Sutton Coldfield Town (1) 1 Moran 10
Att: 887
Redbridge (0) 1 Lincoln 63
Cambridge City (3) 5 Williams 24, J Simpson 29, R Simpson 44, Langston 83, Scott pen 90
Att: 154
Slough Town (2) 4 Harris 11, Hodges 17, 52, Spencer 73
Hendon (0) 3 Seargent 48, Pickett 70, Frendo 90
Att: 405.
Southport (0) 2 Fearns 62
Hucknall Town (2) 2 Ricketts 18, Bacon 22
Att: 770
St Albans City (0) 0
Havant & Waterlooville (1) 1 Holdsworth 37
Att: 373
Stamford (2) 3 Duffy 33, pen 36
Willenhall Town (0) 0
Att: 280
Sutton United (0) 0
Grays Athletic (0) 2 Oli 56, Griffiths 71
Att: 544
Tamworth (2) 5 Cooper 33, Ebdon 42, 50, pen 64, N Smith 84
Gresley Rovers (0) 0
Att: 1,283
Tonbridge Angels (1) 1 Dolby 11
Walton & Hersham (0) 1 Riddle 80
Att: 543
Vauxhall Motors (1) 1 A Wright 32
Lancaster City (1) 1 Sullivan 36
Att: 235
Woking (1) 1 Louis 23
Stevenage Borough (0) 0
Att: 2,139
Yate Town (0) 1 Sims 60
Hayes (1) 1 Knight 36
Att: 310