It was seemingly going all the way of Murray after first set, but Johnson was cunning and got himself back into the match. The American endured a good spell in the middle of the second set where he earned break points on the Murray serve, but he could not convert.
Instead it would be the Brit who made the breakthrough in the final game of the match. Murray applied the pressure at 5-4 and sealed his fourth break of the match to claim victory.
Murray will now face another American in the 6ft10 Isner, who defeated David Goffin 6-3 7-6(5) earlier in the day.
Elsewhere, world number one Novak Djokovic made light work of Martin Klizan for the loss of only three games. The 10 time grand slam champion never got out of first game, but too many errors from the Slovakian meant that it was an easy 6-2, 6-1 victory for the Serb.
Djokovic will play Feliciano Lopez in the third round. The squeezed through a tight three setter against Dominic Thiem (4) 6-7, 7-6(6), 6-3.
Match of the day came last on Stadium court as Rafael Nadal survived the giant test of Ivo Karlovic 7-5, (4)6-7, 7-6(4). Karlovic earned an early break and served for the first set but Nadal was able to muster up consecutive breaks to take the first set. There would be no more breaks of serve in the rest of the match as the pair exchanged tiebreaks. A crucial double fault ended up costing the Croatian in the end.
The Spaniard will face Milos Raonic next after the Canadian defeated Roberto Bautista Agut 6-4, (3)6-7, 7-6(5).
Kei Nishikori was successful in the first match on Stadium court, coming from a set down to defeat Nick Kyrgios 1-6, 6-4, 6-4.
Stan Wawrinka had no trouble dispatching Viktor Troicki 7-6(3), 6-3. Fifth seed Tomas Berdych defeated Jack Sock 7-6(3), 4-6, 6-4, and there were also wins for Kevin Anderson, Richard Gasquet, Bernard Tomic and Gilles Simon.
Murray will face Isner at 6:30AM BST on Thursday morning.
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